Calculus Examples

$y = \sqrt{2 x^{3} + 3 x + 10}$

Step 1

Find the first derivative.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x^{3} + 3 x + 10}$ as ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 2 x^{3} + 3 x + 10$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $2 x^{3} + 3 x + 10$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.2.3

Replace all occurrences of $u$ with $2 x^{3} + 3 x + 10$ .

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1 - 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{- 1}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.7.2

Combine $\frac{1}{2}$ and ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ .

$\frac{{(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.7.3

Move ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} \frac{d}{d x} [2 x^{3} + 3 x + 10]$

Step 1.8

By the Sum Rule, the derivative of $2 x^{3} + 3 x + 10$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10])$

Step 1.9

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10])$

Step 1.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (2 (3 x^{2}) + \frac{d}{d x} [3 x] + \frac{d}{d x} [10])$

Step 1.11

Multiply $3$ by $2$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [10])$

Step 1.12

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [10])$

Step 1.13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 \cdot 1 + \frac{d}{d x} [10])$

Step 1.14

Multiply $3$ by $1$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 + \frac{d}{d x} [10])$

Step 1.15

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 + 0)$

Step 1.16

Add $6 x^{2} + 3$ and $0$ .

$\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)$

Step 1.17

Simplify.

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Step 1.17.1

Reorder the factors of $\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)$ .

$(6 x^{2} + 3) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$

Step 1.17.2

Multiply $6 x^{2} + 3$ by $\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ .

$\frac{6 x^{2} + 3}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$

Step 1.17.3

Factor $3$ out of $6 x^{2} + 3$ .

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Step 1.17.3.1

Factor $3$ out of $6 x^{2}$ .

$\frac{3 (2 x^{2}) + 3}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$

Step 1.17.3.2

Factor $3$ out of $3$ .

$\frac{3 (2 x^{2}) + 3 (1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$

Step 1.17.3.3

Factor $3$ out of $3 (2 x^{2}) + 3 (1)$ .

$f' (x) = \frac{3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [\frac{2 x^{2} + 1}{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}]$ .

$\frac{3}{2} \frac{d}{d x} [\frac{2 x^{2} + 1}{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}]$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 2 x^{2} + 1$ and $g (x) = {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{2} + 1] - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{{({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}^{2}}$

Step 2.3

Multiply the exponents in ${({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{2} + 1] - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

Step 2.3.2.2

Rewrite the expression.

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{2} + 1] - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{{(2 x^{3} + 3 x + 10)}^{1}}$

Step 2.4

Simplify.

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{2} + 1] - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Step 2.5

Differentiate.

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Step 2.5.1

By the Sum Rule, the derivative of $2 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [1]) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Step 2.5.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{2}$ with respect to $x$ is $2 \frac{d}{d x} [x^{2}]$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Step 2.5.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x) + \frac{d}{d x} [1]) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Step 2.5.4

Multiply $2$ by $2$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (4 x + \frac{d}{d x} [1]) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Step 2.5.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (4 x + 0) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Step 2.5.6

Simplify the expression.

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Step 2.5.6.1

Add $4 x$ and $0$ .

$\frac{3}{2} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (4 x) - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Step 2.5.6.2

Move $4$ to the left of ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3}{2} \cdot \frac{4 \cdot {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}]}{2 x^{3} + 3 x + 10}$

Step 2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 2 x^{3} + 3 x + 10$ .

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Step 2.6.1

To apply the Chain Rule, set $u$ as $2 x^{3} + 3 x + 10$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.6.3

Replace all occurrences of $u$ with $2 x^{3} + 3 x + 10$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.8

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.9

Combine the numerators over the common denominator.

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1 - 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.10.2

Subtract $2$ from $1$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{\frac{- 1}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.11

Combine fractions.

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Step 2.11.1

Move the negative in front of the fraction.

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2} {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.11.2

Combine $\frac{1}{2}$ and ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{{(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.11.3

Move ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{2 x^{3} + 3 x + 10}$

Step 2.12

By the Sum Rule, the derivative of $2 x^{3} + 3 x + 10$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Step 2.13

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Step 2.14

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (2 (3 x^{2}) + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Step 2.15

Multiply $3$ by $2$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Step 2.16

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Step 2.17

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 \cdot 1 + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Step 2.18

Multiply $3$ by $1$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 + \frac{d}{d x} [10]))}{2 x^{3} + 3 x + 10}$

Step 2.19

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3 + 0))}{2 x^{3} + 3 x + 10}$

Step 2.20

Combine fractions.

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Step 2.20.1

Add $6 x^{2} + 3$ and $0$ .

$\frac{3}{2} \cdot \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 x^{3} + 3 x + 10}$

Step 2.20.2

Multiply $\frac{3}{2}$ by $\frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 x^{3} + 3 x + 10}$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - (2 x^{2} + 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)))}{2 (2 x^{3} + 3 x + 10)}$

Step 2.21

Simplify.

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Step 2.21.1

Apply the distributive property.

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + (- (2 x^{2}) - 1 \cdot 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)))}{2 (2 x^{3} + 3 x + 10)}$

Step 2.21.2

Apply the distributive property.

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 3 ((- (2 x^{2}) - 1 \cdot 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)))}{2 (2 x^{3} + 3 x + 10)}$

Step 2.21.3

Apply the distributive property.

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 3 ((- (2 x^{2}) - 1 \cdot 1) (\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4

Simplify the numerator.

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Step 2.21.4.1

Factor $3$ out of $3 \cdot 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + 3 (- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)$ .

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Step 2.21.4.1.1

Factor $3$ out of $3 \cdot 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 3 (- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.1.2

Factor $3$ out of $3 (- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3)$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 3 ((- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.1.3

Factor $3$ out of $3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 3 ((- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + (- 1 \cdot 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.2

Simplify each term.

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Step 2.21.4.2.1

Multiply $- 1$ by $2$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + (- 2 x^{2} - 1 \cdot 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.2.2

Multiply $- 1$ by $1$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + (- 2 x^{2} - 1) \frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.3

Multiply $- 2 x^{2} - 1$ by $\frac{1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 2 x^{2} - 1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} (6 x^{2} + 3))}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.4

Multiply $\frac{- 2 x^{2} - 1}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ by $6 x^{2} + 3$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} - 1) (6 x^{2} + 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5

Simplify the numerator.

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Step 2.21.4.5.1

Factor $3$ out of $6 x^{2} + 3$ .

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Step 2.21.4.5.1.1

Factor $3$ out of $6 x^{2}$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} - 1) (3 (2 x^{2}) + 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.1.2

Factor $3$ out of $3$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} - 1) (3 (2 x^{2}) + 3 (1))}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.1.3

Factor $3$ out of $3 (2 x^{2}) + 3 (1)$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} - 1) (3 (2 x^{2} + 1))}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} - 1) \cdot 3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.2

Combine exponents.

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Step 2.21.4.5.2.1

Factor $- 1$ out of $- 2 x^{2}$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- (2 x^{2}) - 1) \cdot 3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.2.2

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{(- (2 x^{2}) - 1 (1)) \cdot 3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.2.3

Factor $- 1$ out of $- (2 x^{2}) - 1 (1)$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- (2 x^{2} + 1) \cdot 3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.2.4

Rewrite $- (2 x^{2} + 1)$ as $- 1 (2 x^{2} + 1)$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 1 (2 x^{2} + 1) \cdot 3 (2 x^{2} + 1)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.2.5

Raise $2 x^{2} + 1$ to the power of $1$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 1 \cdot 3 ({(2 x^{2} + 1)}^{1} (2 x^{2} + 1))}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.2.6

Raise $2 x^{2} + 1$ to the power of $1$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 1 \cdot 3 ({(2 x^{2} + 1)}^{1} {(2 x^{2} + 1)}^{1})}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 1 \cdot 3 {(2 x^{2} + 1)}^{1 + 1}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.2.8

Add $1$ and $1$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 1 \cdot 3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.5.2.9

Multiply $- 1$ by $3$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + \frac{- 3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.6

Move the negative in front of the fraction.

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x - \frac{3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.7

To write $4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x$ as a fraction with a common denominator, multiply by $\frac{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ .

$\frac{3 (4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot \frac{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} - \frac{3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.8

Combine $4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x$ and $\frac{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ .

$\frac{3 (\frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x (2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} - \frac{3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}})}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.9

Combine the numerators over the common denominator.

$\frac{3 \frac{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x (2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) - 3 {(2 x^{2} + 1)}^{2}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.10

Reorder terms.

$\frac{3 \frac{- 3 {(2 x^{2} + 1)}^{2} + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11

Rewrite $\frac{- 3 {(2 x^{2} + 1)}^{2} + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ in a factored form.

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Step 2.21.4.11.1

Rewrite ${(2 x^{2} + 1)}^{2}$ as $(2 x^{2} + 1) (2 x^{2} + 1)$ .

$\frac{3 \frac{- 3 ((2 x^{2} + 1) (2 x^{2} + 1)) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.2

Expand $(2 x^{2} + 1) (2 x^{2} + 1)$ using the FOIL Method.

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Step 2.21.4.11.2.1

Apply the distributive property.

$\frac{3 \frac{- 3 (2 x^{2} (2 x^{2} + 1) + 1 (2 x^{2} + 1)) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.2.2

Apply the distributive property.

$\frac{3 \frac{- 3 (2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2} + 1)) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.2.3

Apply the distributive property.

$\frac{3 \frac{- 3 (2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.3

Simplify and combine like terms.

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Step 2.21.4.11.3.1

Simplify each term.

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Step 2.21.4.11.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{3 \frac{- 3 (2 \cdot 2 x^{2} x^{2} + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.3.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.21.4.11.3.1.2.1

Move $x^{2}$ .

$\frac{3 \frac{- 3 (2 \cdot 2 (x^{2} x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 \frac{- 3 (2 \cdot 2 x^{2 + 2} + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.3.1.2.3

Add $2$ and $2$ .

$\frac{3 \frac{- 3 (2 \cdot 2 x^{4} + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.3.1.3

Multiply $2$ by $2$ .

$\frac{3 \frac{- 3 (4 x^{4} + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.3.1.4

Multiply $2$ by $1$ .

$\frac{3 \frac{- 3 (4 x^{4} + 2 x^{2} + 1 (2 x^{2}) + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.3.1.5

Multiply $2 x^{2}$ by $1$ .

$\frac{3 \frac{- 3 (4 x^{4} + 2 x^{2} + 2 x^{2} + 1 \cdot 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.3.1.6

Multiply $1$ by $1$ .

$\frac{3 \frac{- 3 (4 x^{4} + 2 x^{2} + 2 x^{2} + 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.3.2

Add $2 x^{2}$ and $2 x^{2}$ .

$\frac{3 \frac{- 3 (4 x^{4} + 4 x^{2} + 1) + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.4

Apply the distributive property.

$\frac{3 \frac{- 3 (4 x^{4}) - 3 (4 x^{2}) - 3 \cdot 1 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.5

Simplify.

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Step 2.21.4.11.5.1

Multiply $4$ by $- 3$ .

$\frac{3 \frac{- 12 x^{4} - 3 (4 x^{2}) - 3 \cdot 1 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.5.2

Multiply $4$ by $- 3$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 \cdot 1 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.5.3

Multiply $- 3$ by $1$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.6

Multiply ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ by ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.21.4.11.6.1

Move ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2} + \frac{1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.6.3

Combine the numerators over the common denominator.

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{1 + 1}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.6.4

Add $1$ and $1$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.6.5

Divide $2$ by $2$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{1}}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.7

Simplify $2 \cdot 4 x {(2 x^{3} + 3 x + 10)}^{1}$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 2 \cdot 4 x (2 x^{3} + 3 x + 10)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.8

Multiply $2$ by $4$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 x (2 x^{3} + 3 x + 10)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.9

Apply the distributive property.

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 x (2 x^{3}) + 8 x (3 x) + 8 x \cdot 10}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.10

Simplify.

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Step 2.21.4.11.10.1

Rewrite using the commutative property of multiplication.

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 x \cdot x^{3} + 8 x (3 x) + 8 x \cdot 10}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.10.2

Rewrite using the commutative property of multiplication.

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 x \cdot x^{3} + 8 \cdot 3 x \cdot x + 8 x \cdot 10}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.10.3

Multiply $10$ by $8$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 x \cdot x^{3} + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.11

Simplify each term.

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Step 2.21.4.11.11.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.21.4.11.11.1.1

Move $x^{3}$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 (x^{3} x) + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.11.1.2

Multiply $x^{3}$ by $x$ .

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Step 2.21.4.11.11.1.2.1

Raise $x$ to the power of $1$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 (x^{3} x^{1}) + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.11.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 x^{3 + 1} + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.11.1.3

Add $3$ and $1$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 8 \cdot 2 x^{4} + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.11.2

Multiply $8$ by $2$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 16 x^{4} + 8 \cdot 3 x \cdot x + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.11.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.21.4.11.11.3.1

Move $x$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 16 x^{4} + 8 \cdot 3 (x \cdot x) + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.11.3.2

Multiply $x$ by $x$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 16 x^{4} + 8 \cdot 3 x^{2} + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.11.4

Multiply $8$ by $3$ .

$\frac{3 \frac{- 12 x^{4} - 12 x^{2} - 3 + 16 x^{4} + 24 x^{2} + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.12

Add $- 12 x^{4}$ and $16 x^{4}$ .

$\frac{3 \frac{4 x^{4} - 12 x^{2} - 3 + 24 x^{2} + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.13

Add $- 12 x^{2}$ and $24 x^{2}$ .

$\frac{3 \frac{4 x^{4} + 12 x^{2} - 3 + 80 x}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.4.11.14

Reorder terms.

$\frac{3 \frac{4 x^{4} + 12 x^{2} + 80 x - 3}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.5

Combine terms.

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Step 2.21.5.1

Combine $3$ and $\frac{4 x^{4} + 12 x^{2} + 80 x - 3}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ .

$\frac{\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{2 (2 x^{3}) + 2 (3 x) + 2 \cdot 10}$

Step 2.21.5.2

Multiply $2$ by $2$ .

$\frac{\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{4 x^{3} + 2 (3 x) + 2 \cdot 10}$

Step 2.21.5.3

Multiply $3$ by $2$ .

$\frac{\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{4 x^{3} + 6 x + 2 \cdot 10}$

Step 2.21.5.4

Multiply $2$ by $10$ .

$\frac{\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{4 x^{3} + 6 x + 20}$

Step 2.21.5.5

Rewrite $\frac{\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}}{4 x^{3} + 6 x + 20}$ as a product.

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}} \cdot \frac{1}{4 x^{3} + 6 x + 20}$

Step 2.21.5.6

Multiply $\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}$ by $\frac{1}{4 x^{3} + 6 x + 20}$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (4 x^{3} + 6 x + 20)}$

Step 2.21.6

Simplify the denominator.

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Step 2.21.6.1

Factor $2$ out of $4 x^{3} + 6 x + 20$ .

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Step 2.21.6.1.1

Factor $2$ out of $4 x^{3}$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x^{3}) + 6 x + 20)}$

Step 2.21.6.1.2

Factor $2$ out of $6 x$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x^{3}) + 2 (3 x) + 20)}$

Step 2.21.6.1.3

Factor $2$ out of $20$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x^{3}) + 2 (3 x) + 2 (10))}$

Step 2.21.6.1.4

Factor $2$ out of $2 (2 x^{3}) + 2 (3 x)$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x^{3} + 3 x) + 2 (10))}$

Step 2.21.6.1.5

Factor $2$ out of $2 (2 x^{3} + 3 x) + 2 (10)$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 (2 x^{3} + 3 x + 10))}$

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 (2 x^{3} + 3 x + 10)}$

Step 2.21.6.2

Combine exponents.

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Step 2.21.6.2.1

Multiply $2$ by $2$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 x^{3} + 3 x + 10)}$

Step 2.21.6.2.2

Raise $2 x^{3} + 3 x + 10$ to the power of $1$ .

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 ({(2 x^{3} + 3 x + 10)}^{1} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}$

Step 2.21.6.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{1 + \frac{1}{2}}}$

Step 2.21.6.2.4

Write $1$ as a fraction with a common denominator.

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 2.21.6.2.5

Combine the numerators over the common denominator.

$\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{\frac{2 + 1}{2}}}$

Step 2.21.6.2.6

Add $2$ and $1$ .

$f'' (x) = \frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}}$

Step 3

Find the third derivative.

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Step 3.1

Since $\frac{3}{4}$ is constant with respect to $x$ , the derivative of $\frac{3 (4 x^{4} + 12 x^{2} + 80 x - 3)}{4 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}}$ with respect to $x$ is $\frac{3}{4} \frac{d}{d x} [\frac{4 x^{4} + 12 x^{2} + 80 x - 3}{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}}]$ .

$\frac{3}{4} \frac{d}{d x} [\frac{4 x^{4} + 12 x^{2} + 80 x - 3}{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}}]$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 4 x^{4} + 12 x^{2} + 80 x - 3$ and $g (x) = {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \frac{d}{d x} [4 x^{4} + 12 x^{2} + 80 x - 3] - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}})}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in ${({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}})}^{2}$ .

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Step 3.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

Step 3.3.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.2.1

Cancel the common factor.

Step 3.3.1.2.2

Rewrite the expression.

Step 3.3.2

By the Sum Rule, the derivative of $4 x^{4} + 12 x^{2} + 80 x - 3$ with respect to $x$ is $\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.3

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{4}$ with respect to $x$ is $4 \frac{d}{d x} [x^{4}]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (4 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (4 (4 x^{3}) + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.5

Multiply $4$ by $4$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.6

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 12 (2 x) + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.8

Multiply $2$ by $12$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + \frac{d}{d x} [80 x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.9

Since $80$ is constant with respect to $x$ , the derivative of $80 x$ with respect to $x$ is $80 \frac{d}{d x} [x]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80 \cdot 1 + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.11

Multiply $80$ by $1$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80 + \frac{d}{d x} [- 3]) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.12

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80 + 0) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.3.13

Add $16 x^{3} + 24 x + 80$ and $0$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) \frac{d}{d x} [{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}]}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = 2 x^{3} + 3 x + 10$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $2 x^{3} + 3 x + 10$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{d}{d u} [u^{\frac{3}{2}}] \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.4.3

Replace all occurrences of $u$ with $2 x^{3} + 3 x + 10$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2} - 1} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.6

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.7

Combine the numerators over the common denominator.

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $- 1$ by $2$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{3 - 2}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.8.2

Subtract $2$ from $3$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3}{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.9

Combine $\frac{3}{2}$ and ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} \frac{d}{d x} [2 x^{3} + 3 x + 10])}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.10

By the Sum Rule, the derivative of $2 x^{3} + 3 x + 10$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.11

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.12

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (2 (3 x^{2}) + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.13

Multiply $3$ by $2$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.14

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.15

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3 \cdot 1 + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.16

Multiply $3$ by $1$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3 + \frac{d}{d x} [10]))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.17

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3 + 0))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.18

Combine fractions.

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Step 3.18.1

Add $6 x^{2} + 3$ and $0$ .

$\frac{3}{4} \cdot \frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{{(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.18.2

Multiply $\frac{3}{4}$ by $\frac{{(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{{(2 x^{3} + 3 x + 10)}^{3}}$ .

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) - (4 x^{4} + 12 x^{2} + 80 x - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19

Simplify.

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Step 3.19.1

Apply the distributive property.

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) + (- (4 x^{4}) - (12 x^{2}) - (80 x) - - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.2

Apply the distributive property.

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80)) + 3 ((- (4 x^{4}) - (12 x^{2}) - (80 x) - - 3) (\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3

Simplify the numerator.

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Step 3.19.3.1

Factor $3$ out of $3 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) + 3 (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)$ .

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Step 3.19.3.1.1

Factor $3$ out of $3 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80)$ .

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80)) + 3 (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.1.2

Factor $3$ out of $3 (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3)$ .

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80)) + 3 ((- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.1.3

Factor $3$ out of $3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80)) + 3 ((- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))$ .

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3} + 24 x + 80) + (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.2

Apply the distributive property.

$\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (16 x^{3}) + {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (24 x) + {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 80 + (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.3

Simplify.

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Step 3.19.3.3.1

Rewrite using the commutative property of multiplication.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} (24 x) + {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 80 + (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.3.2

Rewrite using the commutative property of multiplication.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 80 + (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.3.3

Move $80$ to the left of ${(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 \cdot {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 1 \cdot 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.4

Simplify each term.

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Step 3.19.3.4.1

Multiply $- 1$ by $4$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 4 x^{4} - 1 \cdot 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.4.2

Multiply $- 1$ by $12$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 4 x^{4} - 12 x^{2} - 1 \cdot 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.4.3

Multiply $- 1$ by $80$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 4 x^{4} - 12 x^{2} - 80 x - - 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.4.4

Multiply $- 1$ by $- 3$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 4 x^{4} - 12 x^{2} - 80 x + 3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.5

Apply the distributive property.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 4 x^{4} \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 12 x^{2} \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6

Simplify.

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Step 3.19.3.6.1

Cancel the common factor of $2$ .

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Step 3.19.3.6.1.1

Factor $2$ out of $- 4 x^{4}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (2 (- 2 x^{4}) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 12 x^{2} \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.1.2

Cancel the common factor.

Step 3.19.3.6.1.3

Rewrite the expression.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 2 x^{4} (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) - 12 x^{2} \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.2

Multiply $3$ by $- 2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 12 x^{2} \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.3

Cancel the common factor of $2$ .

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Step 3.19.3.6.3.1

Factor $2$ out of $- 12 x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 2 (- 6 x^{2}) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.3.2

Cancel the common factor.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 2 (- 6 x^{2}) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.3.3

Rewrite the expression.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 6 x^{2} (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.4

Multiply $3$ by $- 6$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 80 x \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.5

Cancel the common factor of $2$ .

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Step 3.19.3.6.5.1

Factor $2$ out of $- 80 x$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 2 (- 40 x) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.5.2

Cancel the common factor.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 2 (- 40 x) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.5.3

Rewrite the expression.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 40 x (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.6

Multiply $3$ by $- 40$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.7

Multiply $3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}$ .

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Step 3.19.3.6.7.1

Combine $3$ and $\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.6.7.2

Multiply $3$ by $3$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.7

To write $- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.8

Combine $- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.9

Combine the numerators over the common denominator.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.10

Simplify the numerator.

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Step 3.19.3.10.1

Factor $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 6 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

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Step 3.19.3.10.1.1

Reorder the expression.

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Step 3.19.3.10.1.1.1

Move ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 6 x^{4} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.10.1.1.2

Move $x^{4}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 6 \cdot 2 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.10.1.2

Factor $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 6 \cdot 2 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 2 \cdot 2 x^{4}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.10.1.3

Factor $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 2 \cdot 2 x^{4}) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.10.1.4

Factor $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 2 \cdot 2 x^{4}) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 2 \cdot 2 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.10.2

Multiply $- 2$ by $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.11

To write $- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} - 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.12

Combine $- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2}{2} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.13

Combine the numerators over the common denominator.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.14

Simplify the numerator.

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Step 3.19.3.14.1

Factor $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 18 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)$ .

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Step 3.19.3.14.1.1

Reorder the expression.

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Step 3.19.3.14.1.1.1

Move ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 18 x^{2} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.14.1.1.2

Move $x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{- 18 \cdot 2 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.14.1.2

Factor $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 18 \cdot 2 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 6 \cdot 2 x^{2}) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.14.1.3

Factor $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 6 \cdot 2 x^{2}) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 6 \cdot 2 x^{2} - 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.14.2

Multiply $- 6$ by $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 x^{2} - 4 x^{4} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.14.3

Reorder terms.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.15

To write $- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.16

Combine $- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (\frac{- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2}{2} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2}) (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.17

Combine the numerators over the common denominator.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.18

Simplify the numerator.

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Step 3.19.3.18.1

Factor $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 120 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 2 + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)$ .

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Step 3.19.3.18.1.1

Reorder the expression.

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Step 3.19.3.18.1.1.1

Move ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{- 120 x \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.18.1.1.2

Move $x$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{- 120 \cdot 2 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.18.1.2

Factor $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 120 \cdot 2 x {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 40 \cdot 2 x) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.18.1.3

Factor $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 40 \cdot 2 x) + 3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 40 \cdot 2 x - 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.18.2

Multiply $- 40$ by $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 x - 4 x^{4} - 12 x^{2} + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.18.3

Reorder terms.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} (6 x^{2} + 3))}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.19

Apply the distributive property.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} (6 x^{2}) + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} \cdot 3)}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.20

Rewrite using the commutative property of multiplication.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 6 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} x^{2} + \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} \cdot 3)}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.21

Multiply $\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} \cdot 3$ .

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Step 3.19.3.21.1

Combine $\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2}$ and $3$ .

Step 3.19.3.21.2

Multiply $3$ by $3$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 6 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22

Simplify each term.

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Step 3.19.3.22.1

Cancel the common factor of $2$ .

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Step 3.19.3.22.1.1

Factor $2$ out of $6$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 (3) \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.1.2

Cancel the common factor.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 3 \frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.1.3

Rewrite the expression.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 3 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.2

Multiply $3$ by $3$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.3

Apply the distributive property.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 x^{2}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 x) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 3) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.4

Simplify.

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Step 3.19.3.22.4.1

Rewrite using the commutative property of multiplication.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 (- 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 x^{2}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 x) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 3) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.4.2

Rewrite using the commutative property of multiplication.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 (- 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 12 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 x) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 3) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.4.3

Rewrite using the commutative property of multiplication.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 (- 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 12 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 80 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} \cdot 3) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.4.4

Move $3$ to the left of ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 (- 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 12 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 80 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + 3 \cdot {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.5

Apply the distributive property.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (9 (- 4 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}) + 9 (- 12 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}) + 9 (- 80 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 9 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.6

Simplify.

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Step 3.19.3.22.6.1

Multiply $- 4$ by $9$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 36 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}) + 9 (- 12 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}) + 9 (- 80 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 9 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.6.2

Multiply $- 12$ by $9$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 36 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}) - 108 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}) + 9 (- 80 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 9 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.6.3

Multiply $- 80$ by $9$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 36 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}) - 108 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}) - 720 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 9 (3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}})) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.6.4

Multiply $3$ by $9$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 36 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}) - 108 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}) - 720 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x) + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.7

Remove parentheses.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + (- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}) x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.8

Apply the distributive property.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} x^{2} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.9

Simplify.

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Step 3.19.3.22.9.1

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 3.19.3.22.9.1.1

Move $x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (x^{2} x^{4}) - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.9.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2 + 4} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.9.1.3

Add $2$ and $4$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.9.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.19.3.22.9.2.1

Move $x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (x^{2} x^{2}) - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.9.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2 + 2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.9.2.3

Add $2$ and $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x \cdot x^{2} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.9.3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.19.3.22.9.3.1

Move $x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (x^{2} x) + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.9.3.2

Multiply $x^{2}$ by $x$ .

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Step 3.19.3.22.9.3.2.1

Raise $x$ to the power of $1$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (x^{2} x^{1}) + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.9.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2 + 1} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.22.9.3.3

Add $2$ and $1$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.23

To write $- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.24

Combine $- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6}$ and $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.25

Combine the numerators over the common denominator.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.26

Simplify the numerator.

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Step 3.19.3.26.1

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 36 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{6} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)$ .

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Step 3.19.3.26.1.1

Reorder the expression.

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Step 3.19.3.26.1.1.1

Move ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 36 x^{6} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.26.1.1.2

Move $x^{6}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 36 \cdot 2 x^{6} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.26.1.2

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 36 \cdot 2 x^{6} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 \cdot 2 x^{6}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.26.1.3

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 \cdot 2 x^{6}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 x^{4} - 12 x^{2} - 80 x + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 4 \cdot 2 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.26.2

Multiply $- 4$ by $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.27

To write $- 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.28

Combine $- 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4}$ and $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.29

Combine the numerators over the common denominator.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.30

Simplify the numerator.

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Step 3.19.3.30.1

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 108 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{4} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)$ .

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Step 3.19.3.30.1.1

Reorder the expression.

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Step 3.19.3.30.1.1.1

Move ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 108 x^{4} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.30.1.1.2

Move $x^{4}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 108 \cdot 2 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.30.1.2

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 108 \cdot 2 x^{4} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 \cdot 2 x^{4}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.30.1.3

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 \cdot 2 x^{4}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 12 \cdot 2 x^{4} - 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.30.2

Multiply $- 12$ by $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 24 x^{4} - 8 x^{6} - 4 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.30.3

Subtract $4 x^{4}$ from $- 24 x^{4}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.31

To write $- 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} - 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.32

Combine $- 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3}$ and $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.33

Combine the numerators over the common denominator.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.34

Simplify the numerator.

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Step 3.19.3.34.1

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 720 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{3} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)$ .

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Step 3.19.3.34.1.1

Reorder the expression.

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Step 3.19.3.34.1.1.1

Move ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 720 x^{3} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.34.1.1.2

Move $x^{3}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{- 720 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.34.1.2

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $- 720 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 \cdot 2 x^{3}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.34.1.3

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 \cdot 2 x^{3}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 80 \cdot 2 x^{3} - 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.34.2

Multiply $- 80$ by $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 160 x^{3} - 8 x^{6} - 28 x^{4} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.34.3

Reorder terms.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.35

To write $27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.36

Combine $27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2}$ and $\frac{2}{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.37

Combine the numerators over the common denominator.

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.38

Simplify the numerator.

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Step 3.19.3.38.1

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $27 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} x^{2} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)$ .

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Step 3.19.3.38.1.1

Reorder the expression.

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Step 3.19.3.38.1.1.1

Move ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{27 x^{2} \cdot 2 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.38.1.1.2

Move $x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{27 \cdot 2 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.38.1.2

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $27 \cdot 2 x^{2} {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (3 \cdot 2 x^{2}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.38.1.3

Factor $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (3 \cdot 2 x^{2}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (3 \cdot 2 x^{2} - 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.38.2

Multiply $3$ by $2$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (6 x^{2} - 8 x^{6} - 28 x^{4} - 160 x^{3} - 12 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.38.3

Subtract $12 x^{2}$ from $6 x^{2}$ .

$\frac{3 (16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} + 24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.39

To write $16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} \cdot \frac{2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.40

Combine $16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3}$ and $\frac{2}{2}$ .

$\frac{3 (24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} \cdot 2}{2} + \frac{9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.41

Combine the numerators over the common denominator.

$\frac{3 (24 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{16 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} x^{3} \cdot 2 + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.42

Move $x^{3}$ .

$\frac{3 (80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.43

To write $80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.44

Combine $80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$\frac{3 (24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2}{2} + \frac{16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.45

Combine the numerators over the common denominator.

$\frac{3 (24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + \frac{80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2 + 16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.46

To write $24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2 + 16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.47

Combine $24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$\frac{3 (\frac{24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2}{2} + \frac{80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2 + 16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2})}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.48

Combine the numerators over the common denominator.

$\frac{3 \frac{24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2 + 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} \cdot 2 + 16 \cdot 2 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.49

Reorder terms.

$\frac{3 \frac{2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50

Rewrite $\frac{2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}$ in a factored form.

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Step 3.19.3.50.1

Factor ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)$ .

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Step 3.19.3.50.1.1

Factor ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.1.2

Factor ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}} + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.1.3

Factor ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{3}{2}}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + 9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.1.4

Factor ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of $9 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3)$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.1.5

Factor ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}})$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.1.6

Factor ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}})$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.1.7

Factor ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ out of ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}}) + {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (2 \cdot 24 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.2

Multiply $2$ by $24$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 x {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.3

Divide $2$ by $2$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 x {(2 x^{3} + 3 x + 10)}^{1} + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.4

Simplify.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 x (2 x^{3} + 3 x + 10) + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.5

Apply the distributive property.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 x (2 x^{3}) + 48 x (3 x) + 48 x \cdot 10 + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.6

Simplify.

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Step 3.19.3.50.6.1

Rewrite using the commutative property of multiplication.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 x \cdot x^{3} + 48 x (3 x) + 48 x \cdot 10 + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.6.2

Rewrite using the commutative property of multiplication.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 x \cdot x^{3} + 48 \cdot 3 x \cdot x + 48 x \cdot 10 + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.6.3

Multiply $10$ by $48$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 x \cdot x^{3} + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.7

Simplify each term.

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Step 3.19.3.50.7.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 3.19.3.50.7.1.1

Move $x^{3}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 (x^{3} x) + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.7.1.2

Multiply $x^{3}$ by $x$ .

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Step 3.19.3.50.7.1.2.1

Raise $x$ to the power of $1$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 (x^{3} x^{1}) + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.7.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 x^{3 + 1} + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.7.1.3

Add $3$ and $1$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (48 \cdot 2 x^{4} + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.7.2

Multiply $48$ by $2$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 48 \cdot 3 x \cdot x + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.7.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.19.3.50.7.3.1

Move $x$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 48 \cdot 3 (x \cdot x) + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.7.3.2

Multiply $x$ by $x$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 48 \cdot 3 x^{2} + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.7.4

Multiply $48$ by $3$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 2 \cdot 80 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.8

Multiply $2$ by $80$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 160 {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.9

Divide $2$ by $2$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 160 {(2 x^{3} + 3 x + 10)}^{1} + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.10

Simplify.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 160 (2 x^{3} + 3 x + 10) + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.11

Apply the distributive property.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 160 (2 x^{3}) + 160 (3 x) + 160 \cdot 10 + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.12

Simplify.

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Step 3.19.3.50.12.1

Multiply $2$ by $160$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 160 (3 x) + 160 \cdot 10 + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.12.2

Multiply $3$ by $160$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 160 \cdot 10 + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.12.3

Multiply $160$ by $10$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 2 \cdot 16 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.13

Multiply $2$ by $16$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 x^{3} {(2 x^{3} + 3 x + 10)}^{\frac{2}{2}} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.14

Divide $2$ by $2$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 x^{3} {(2 x^{3} + 3 x + 10)}^{1} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.15

Simplify.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 x^{3} (2 x^{3} + 3 x + 10) + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.16

Apply the distributive property.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 x^{3} (2 x^{3}) + 32 x^{3} (3 x) + 32 x^{3} \cdot 10 + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.17

Simplify.

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Step 3.19.3.50.17.1

Rewrite using the commutative property of multiplication.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 x^{3} x^{3} + 32 x^{3} (3 x) + 32 x^{3} \cdot 10 + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.17.2

Rewrite using the commutative property of multiplication.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 x^{3} x^{3} + 32 \cdot 3 x^{3} x + 32 x^{3} \cdot 10 + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.17.3

Multiply $10$ by $32$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 x^{3} x^{3} + 32 \cdot 3 x^{3} x + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.18

Simplify each term.

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Step 3.19.3.50.18.1

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 3.19.3.50.18.1.1

Move $x^{3}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 (x^{3} x^{3}) + 32 \cdot 3 x^{3} x + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.18.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 x^{3 + 3} + 32 \cdot 3 x^{3} x + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.18.1.3

Add $3$ and $3$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 32 \cdot 2 x^{6} + 32 \cdot 3 x^{3} x + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.18.2

Multiply $32$ by $2$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 32 \cdot 3 x^{3} x + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.18.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 3.19.3.50.18.3.1

Move $x$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 32 \cdot 3 (x \cdot x^{3}) + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.18.3.2

Multiply $x$ by $x^{3}$ .

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Step 3.19.3.50.18.3.2.1

Raise $x$ to the power of $1$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 32 \cdot 3 (x^{1} x^{3}) + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.18.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 32 \cdot 3 x^{1 + 3} + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.18.3.3

Add $1$ and $3$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 32 \cdot 3 x^{4} + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.18.4

Multiply $32$ by $3$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} + 9 (- 8 x^{6} - 28 x^{4} - 160 x^{3} - 6 x^{2} - 80 x + 3))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.19

Apply the distributive property.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} + 9 (- 8 x^{6}) + 9 (- 28 x^{4}) + 9 (- 160 x^{3}) + 9 (- 6 x^{2}) + 9 (- 80 x) + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.20

Simplify.

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Step 3.19.3.50.20.1

Multiply $- 8$ by $9$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} - 72 x^{6} + 9 (- 28 x^{4}) + 9 (- 160 x^{3}) + 9 (- 6 x^{2}) + 9 (- 80 x) + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.20.2

Multiply $- 28$ by $9$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} - 72 x^{6} - 252 x^{4} + 9 (- 160 x^{3}) + 9 (- 6 x^{2}) + 9 (- 80 x) + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.20.3

Multiply $- 160$ by $9$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} - 72 x^{6} - 252 x^{4} - 1440 x^{3} + 9 (- 6 x^{2}) + 9 (- 80 x) + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.20.4

Multiply $- 6$ by $9$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} - 72 x^{6} - 252 x^{4} - 1440 x^{3} - 54 x^{2} + 9 (- 80 x) + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.20.5

Multiply $- 80$ by $9$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (96 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 96 x^{4} + 320 x^{3} - 72 x^{6} - 252 x^{4} - 1440 x^{3} - 54 x^{2} - 720 x + 9 \cdot 3)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.20.6

Multiply $9$ by $3$ .

Step 3.19.3.50.21

Add $96 x^{4}$ and $96 x^{4}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (192 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 320 x^{3} - 72 x^{6} - 252 x^{4} - 1440 x^{3} - 54 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.22

Subtract $252 x^{4}$ from $192 x^{4}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} + 144 x^{2} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 320 x^{3} - 72 x^{6} - 1440 x^{3} - 54 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.23

Subtract $54 x^{2}$ from $144 x^{2}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} + 480 x + 320 x^{3} + 480 x + 1600 + 64 x^{6} + 320 x^{3} - 72 x^{6} - 1440 x^{3} + 90 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.24

Add $480 x$ and $480 x$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} + 320 x^{3} + 960 x + 1600 + 64 x^{6} + 320 x^{3} - 72 x^{6} - 1440 x^{3} + 90 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.25

Add $320 x^{3}$ and $320 x^{3}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} + 640 x^{3} + 960 x + 1600 + 64 x^{6} - 72 x^{6} - 1440 x^{3} + 90 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.26

Subtract $1440 x^{3}$ from $640 x^{3}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} - 800 x^{3} + 960 x + 1600 + 64 x^{6} - 72 x^{6} + 90 x^{2} - 720 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.27

Subtract $720 x$ from $960 x$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} - 800 x^{3} + 1600 + 64 x^{6} - 72 x^{6} + 90 x^{2} + 240 x + 27)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.28

Add $1600$ and $27$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} - 800 x^{3} + 64 x^{6} - 72 x^{6} + 90 x^{2} + 240 x + 1627)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.29

Subtract $72 x^{6}$ from $64 x^{6}$ .

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 60 x^{4} - 800 x^{3} - 8 x^{6} + 90 x^{2} + 240 x + 1627)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.3.50.30

Reorder terms.

$\frac{3 \frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.4

Combine terms.

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Step 3.19.4.1

Combine $3$ and $\frac{{(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2}$ .

$\frac{\frac{3 ({(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627))}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.4.2

Rewrite $\frac{\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2}}{4 {(2 x^{3} + 3 x + 10)}^{3}}$ as a product.

$\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2} \cdot \frac{1}{4 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.4.3

Multiply $\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2}$ by $\frac{1}{4 {(2 x^{3} + 3 x + 10)}^{3}}$ .

$\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{2 (4 {(2 x^{3} + 3 x + 10)}^{3})}$

Step 3.19.4.4

Multiply $4$ by $2$ .

$\frac{3 {(2 x^{3} + 3 x + 10)}^{\frac{1}{2}} (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{3}}$

Step 3.19.4.5

Move ${(2 x^{3} + 3 x + 10)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{3} {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}}$

Step 3.19.4.6

Multiply ${(2 x^{3} + 3 x + 10)}^{3}$ by ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 3.19.4.6.1

Move ${(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}}$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 ({(2 x^{3} + 3 x + 10)}^{- \frac{1}{2}} {(2 x^{3} + 3 x + 10)}^{3})}$

Step 3.19.4.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2} + 3}}$

Step 3.19.4.6.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2} + 3 \cdot \frac{2}{2}}}$

Step 3.19.4.6.4

Combine $3$ and $\frac{2}{2}$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{- \frac{1}{2} + \frac{3 \cdot 2}{2}}}$

Step 3.19.4.6.5

Combine the numerators over the common denominator.

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{- 1 + 3 \cdot 2}{2}}}$

Step 3.19.4.6.6

Simplify the numerator.

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Step 3.19.4.6.6.1

Multiply $3$ by $2$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{- 1 + 6}{2}}}$

Step 3.19.4.6.6.2

Add $- 1$ and $6$ .

$\frac{3 (- 8 x^{6} - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.5

Factor $- 1$ out of $- 8 x^{6}$ .

$\frac{3 (- (8 x^{6}) - 60 x^{4} - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.6

Factor $- 1$ out of $- 60 x^{4}$ .

$\frac{3 (- (8 x^{6}) - (60 x^{4}) - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.7

Factor $- 1$ out of $- (8 x^{6}) - (60 x^{4})$ .

$\frac{3 (- (8 x^{6} + 60 x^{4}) - 800 x^{3} + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.8

Factor $- 1$ out of $- 800 x^{3}$ .

$\frac{3 (- (8 x^{6} + 60 x^{4}) - (800 x^{3}) + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.9

Factor $- 1$ out of $- (8 x^{6} + 60 x^{4}) - (800 x^{3})$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3}) + 90 x^{2} + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.10

Factor $- 1$ out of $90 x^{2}$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3}) - (- 90 x^{2}) + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.11

Factor $- 1$ out of $- (8 x^{6} + 60 x^{4} + 800 x^{3}) - (- 90 x^{2})$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2}) + 240 x + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.12

Factor $- 1$ out of $240 x$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2}) - (- 240 x) + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.13

Factor $- 1$ out of $- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2}) - (- 240 x)$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x) + 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.14

Rewrite $1627$ as $- 1 (- 1627)$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x) - 1 (- 1627))}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.15

Factor $- 1$ out of $- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x) - 1 (- 1627)$ .

$\frac{3 (- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x - 1627))}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.16

Rewrite $- (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x - 1627)$ as $- 1 (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x - 1627)$ .

$\frac{3 (- 1 (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x - 1627))}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$

Step 3.19.17

Move the negative in front of the fraction.

$f''' (x) = - \frac{3 (8 x^{6} + 60 x^{4} + 800 x^{3} - 90 x^{2} - 240 x - 1627)}{8 {(2 x^{3} + 3 x + 10)}^{\frac{5}{2}}}$