Calculus Examples

$y = {(3 x^{2} + \frac{4}{5} x)}^{5} (\sqrt{8 x + 7})$

Step 1

Combine fractions.

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Step 1.1

Combine $\frac{4}{5}$ and $x$ .

$\frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5} \sqrt{8 x + 7}]$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 x + 7}$ as ${(8 x + 7)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5} {(8 x + 7)}^{\frac{1}{2}}]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(3 x^{2} + \frac{4 x}{5})}^{5}$ and $g (x) = {(8 x + 7)}^{\frac{1}{2}}$ .

${(3 x^{2} + \frac{4 x}{5})}^{5} \frac{d}{d x} [{(8 x + 7)}^{\frac{1}{2}}] + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 3

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 $g (x) = 8 x + 7$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $8 x + 7$ .

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 3.2

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

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 3.3

Replace all occurrences of $u_{1}$ with $8 x + 7$ .

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{1}{2} {(8 x + 7)}^{\frac{1}{2} - 1} \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 4

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

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{1}{2} {(8 x + 7)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 5

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

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{1}{2} {(8 x + 7)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 6

Combine the numerators over the common denominator.

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{1}{2} {(8 x + 7)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{1}{2} {(8 x + 7)}^{\frac{1 - 2}{2}} \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 7.2

Subtract $2$ from $1$ .

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{1}{2} {(8 x + 7)}^{\frac{- 1}{2}} \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{1}{2} {(8 x + 7)}^{- \frac{1}{2}} \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 8.2

Combine $\frac{1}{2}$ and ${(8 x + 7)}^{- \frac{1}{2}}$ .

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{{(8 x + 7)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 8.3

Move ${(8 x + 7)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(3 x^{2} + \frac{4 x}{5})}^{5} (\frac{1}{2 {(8 x + 7)}^{\frac{1}{2}}} \frac{d}{d x} [8 x + 7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 8.4

Combine $\frac{1}{2 {(8 x + 7)}^{\frac{1}{2}}}$ and ${(3 x^{2} + \frac{4 x}{5})}^{5}$ .

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{5}}{2 {(8 x + 7)}^{\frac{1}{2}}} \frac{d}{d x} [8 x + 7] + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 9

By the Sum Rule, the derivative of $8 x + 7$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [7]$ .

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{5}}{2 {(8 x + 7)}^{\frac{1}{2}}} (\frac{d}{d x} [8 x] + \frac{d}{d x} [7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 10

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{5}}{2 {(8 x + 7)}^{\frac{1}{2}}} (8 \frac{d}{d x} [x] + \frac{d}{d x} [7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 11

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

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{5}}{2 {(8 x + 7)}^{\frac{1}{2}}} (8 \cdot 1 + \frac{d}{d x} [7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 12

Multiply $8$ by $1$ .

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{5}}{2 {(8 x + 7)}^{\frac{1}{2}}} (8 + \frac{d}{d x} [7]) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 13

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{5}}{2 {(8 x + 7)}^{\frac{1}{2}}} (8 + 0) + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 14

Simplify terms.

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Step 14.1

Add $8$ and $0$ .

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{5}}{2 {(8 x + 7)}^{\frac{1}{2}}} \cdot 8 + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 14.2

Combine $\frac{{(3 x^{2} + \frac{4 x}{5})}^{5}}{2 {(8 x + 7)}^{\frac{1}{2}}}$ and $8$ .

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{5} \cdot 8}{2 {(8 x + 7)}^{\frac{1}{2}}} + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 14.3

Move $8$ to the left of ${(3 x^{2} + \frac{4 x}{5})}^{5}$ .

$\frac{8 \cdot {(3 x^{2} + \frac{4 x}{5})}^{5}}{2 {(8 x + 7)}^{\frac{1}{2}}} + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 14.4

Factor $2$ out of $8 {(3 x^{2} + \frac{4 x}{5})}^{5}$ .

$\frac{2 (4 {(3 x^{2} + \frac{4 x}{5})}^{5})}{2 {(8 x + 7)}^{\frac{1}{2}}} + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 15

Cancel the common factors.

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Step 15.1

Factor $2$ out of $2 {(8 x + 7)}^{\frac{1}{2}}$ .

$\frac{2 (4 {(3 x^{2} + \frac{4 x}{5})}^{5})}{2 ({(8 x + 7)}^{\frac{1}{2}})} + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 15.2

Cancel the common factor.

$\frac{2 (4 {(3 x^{2} + \frac{4 x}{5})}^{5})}{2 {(8 x + 7)}^{\frac{1}{2}}} + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 15.3

Rewrite the expression.

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + {(8 x + 7)}^{\frac{1}{2}} \frac{d}{d x} [{(3 x^{2} + \frac{4 x}{5})}^{5}]$

Step 16

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = 3 x^{2} + \frac{4 x}{5}$ .

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Step 16.1

To apply the Chain Rule, set $u_{2}$ as $3 x^{2} + \frac{4 x}{5}$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + {(8 x + 7)}^{\frac{1}{2}} (\frac{d}{d u_{2}} [{u_{2}}^{5}] \frac{d}{d x} [3 x^{2} + \frac{4 x}{5}])$

Step 16.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 5$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + {(8 x + 7)}^{\frac{1}{2}} (5 {u_{2}}^{4} \frac{d}{d x} [3 x^{2} + \frac{4 x}{5}])$

Step 16.3

Replace all occurrences of $u_{2}$ with $3 x^{2} + \frac{4 x}{5}$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + {(8 x + 7)}^{\frac{1}{2}} (5 {(3 x^{2} + \frac{4 x}{5})}^{4} \frac{d}{d x} [3 x^{2} + \frac{4 x}{5}])$

Step 17

Differentiate.

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Step 17.1

Move $5$ to the left of ${(8 x + 7)}^{\frac{1}{2}}$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + 5 \cdot {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} \frac{d}{d x} [3 x^{2} + \frac{4 x}{5}]$

Step 17.2

By the Sum Rule, the derivative of $3 x^{2} + \frac{4 x}{5}$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [\frac{4 x}{5}]$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + 5 {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [\frac{4 x}{5}])$

Step 17.3

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

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + 5 {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{4 x}{5}])$

Step 17.4

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

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + 5 {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (3 (2 x) + \frac{d}{d x} [\frac{4 x}{5}])$

Step 17.5

Multiply $2$ by $3$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + 5 {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{d}{d x} [\frac{4 x}{5}])$

Step 17.6

Since $\frac{4}{5}$ is constant with respect to $x$ , the derivative of $\frac{4 x}{5}$ with respect to $x$ is $\frac{4}{5} \frac{d}{d x} [x]$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + 5 {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5} \frac{d}{d x} [x])$

Step 17.7

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

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + 5 {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5} \cdot 1)$

Step 17.8

Multiply $\frac{4}{5}$ by $1$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + 5 {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})$

Step 18

To write $5 {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})$ as a fraction with a common denominator, multiply by $\frac{{(8 x + 7)}^{\frac{1}{2}}}{{(8 x + 7)}^{\frac{1}{2}}}$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5}}{{(8 x + 7)}^{\frac{1}{2}}} + \frac{5 {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5}) {(8 x + 7)}^{\frac{1}{2}}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 19

Combine the numerators over the common denominator.

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5} + 5 {(8 x + 7)}^{\frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5}) {(8 x + 7)}^{\frac{1}{2}}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 20

Multiply ${(8 x + 7)}^{\frac{1}{2}}$ by ${(8 x + 7)}^{\frac{1}{2}}$ by adding the exponents.

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Step 20.1

Move ${(8 x + 7)}^{\frac{1}{2}}$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5} + 5 ({(8 x + 7)}^{\frac{1}{2}} {(8 x + 7)}^{\frac{1}{2}}) {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 20.2

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

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5} + 5 {(8 x + 7)}^{\frac{1}{2} + \frac{1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 20.3

Combine the numerators over the common denominator.

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5} + 5 {(8 x + 7)}^{\frac{1 + 1}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 20.4

Add $1$ and $1$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5} + 5 {(8 x + 7)}^{\frac{2}{2}} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 20.5

Divide $2$ by $2$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5} + 5 {(8 x + 7)}^{1} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 21

Simplify $5 {(8 x + 7)}^{1} {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})$ .

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5} + 5 (8 x + 7) {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22

Simplify.

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Step 22.1

Apply the distributive property.

$\frac{4 {(3 x^{2} + \frac{4 x}{5})}^{5} + (5 (8 x) + 5 \cdot 7) {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2

Simplify the numerator.

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Step 22.2.1

Factor ${(3 x^{2} + \frac{4 x}{5})}^{4}$ out of $4 {(3 x^{2} + \frac{4 x}{5})}^{5} + (5 \cdot 8 x + 5 \cdot 7) {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})$ .

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Step 22.2.1.1

Factor ${(3 x^{2} + \frac{4 x}{5})}^{4}$ out of $4 {(3 x^{2} + \frac{4 x}{5})}^{5}$ .

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5})) + (5 \cdot 8 x + 5 \cdot 7) {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.1.2

Factor ${(3 x^{2} + \frac{4 x}{5})}^{4}$ out of $(5 \cdot 8 x + 5 \cdot 7) {(3 x^{2} + \frac{4 x}{5})}^{4} (6 x + \frac{4}{5})$ .

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5})) + {(3 x^{2} + \frac{4 x}{5})}^{4} ((5 \cdot 8 x + 5 \cdot 7) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.1.3

Factor ${(3 x^{2} + \frac{4 x}{5})}^{4}$ out of ${(3 x^{2} + \frac{4 x}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5})) + {(3 x^{2} + \frac{4 x}{5})}^{4} ((5 \cdot 8 x + 5 \cdot 7) (6 x + \frac{4}{5}))$ .

$\frac{{(3 x^{2} + \frac{4 x}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (5 \cdot 8 x + 5 \cdot 7) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.2

Factor $x$ out of $3 x^{2} + \frac{4 x}{5}$ .

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Step 22.2.2.1

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

$\frac{{(x (3 x) + \frac{4 x}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (5 \cdot 8 x + 5 \cdot 7) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.2.2

Factor $x$ out of $\frac{4 x}{5}$ .

$\frac{{(x (3 x) + x \frac{4}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (5 \cdot 8 x + 5 \cdot 7) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.2.3

Factor $x$ out of $x (3 x) + x \frac{4}{5}$ .

$\frac{{(x (3 x + \frac{4}{5}))}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (5 \cdot 8 x + 5 \cdot 7) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.3

Apply the product rule to $x (3 x + \frac{4}{5})$ .

$\frac{x^{4} {(3 x + \frac{4}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (5 \cdot 8 x + 5 \cdot 7) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.4

Combine exponents.

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Step 22.2.4.1

Multiply $5$ by $8$ .

$\frac{x^{4} {(3 x + \frac{4}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (40 x + 5 \cdot 7) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.4.2

Multiply $5$ by $7$ .

$\frac{x^{4} {(3 x + \frac{4}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.5

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x^{4} {(3 x \cdot \frac{5}{5} + \frac{4}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.6

Combine $3 x$ and $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{3 x \cdot 5}{5} + \frac{4}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.7

Combine the numerators over the common denominator.

$\frac{x^{4} {(\frac{3 x \cdot 5 + 4}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.8

Multiply $5$ by $3$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (4 (3 x^{2} + \frac{4 x}{5}) + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9

Simplify each term.

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Step 22.2.9.1

To write $3 x^{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (4 (3 x^{2} \cdot \frac{5}{5} + \frac{4 x}{5}) + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.2

Combine $3 x^{2}$ and $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (4 (\frac{3 x^{2} \cdot 5}{5} + \frac{4 x}{5}) + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.3

Combine the numerators over the common denominator.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (4 \frac{3 x^{2} \cdot 5 + 4 x}{5} + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.4

Simplify the numerator.

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Step 22.2.9.4.1

Factor $x$ out of $3 x^{2} \cdot 5 + 4 x$ .

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Step 22.2.9.4.1.1

Factor $x$ out of $3 x^{2} \cdot 5$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (4 \frac{x (3 x \cdot 5) + 4 x}{5} + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.4.1.2

Factor $x$ out of $4 x$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (4 \frac{x (3 x \cdot 5) + x \cdot 4}{5} + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.4.1.3

Factor $x$ out of $x (3 x \cdot 5) + x \cdot 4$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (4 \frac{x (3 x \cdot 5 + 4)}{5} + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.4.2

Multiply $5$ by $3$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (4 \frac{x (15 x + 4)}{5} + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.5

Combine $4$ and $\frac{x (15 x + 4)}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 (x (15 x + 4))}{5} + (40 x + 35) (6 x + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.6

To write $6 x$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + (40 x + 35) (6 x \cdot \frac{5}{5} + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.7

Combine $6 x$ and $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + (40 x + 35) (\frac{6 x \cdot 5}{5} + \frac{4}{5}))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.8

Combine the numerators over the common denominator.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + (40 x + 35) \frac{6 x \cdot 5 + 4}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.9

Simplify the numerator.

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Step 22.2.9.9.1

Factor $2$ out of $6 x \cdot 5 + 4$ .

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Step 22.2.9.9.1.1

Factor $2$ out of $6 x \cdot 5$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + (40 x + 35) \frac{2 (3 x \cdot 5) + 4}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.9.1.2

Factor $2$ out of $4$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + (40 x + 35) \frac{2 (3 x \cdot 5) + 2 (2)}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.9.1.3

Factor $2$ out of $2 (3 x \cdot 5) + 2 (2)$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + (40 x + 35) \frac{2 (3 x \cdot 5 + 2)}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.9.2

Multiply $5$ by $3$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + (40 x + 35) \frac{2 (15 x + 2)}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.10

Apply the distributive property.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 40 x \frac{2 (15 x + 2)}{5} + 35 \frac{2 (15 x + 2)}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.11

Cancel the common factor of $5$ .

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Step 22.2.9.11.1

Factor $5$ out of $40 x$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 5 (8 x) \frac{2 (15 x + 2)}{5} + 35 \frac{2 (15 x + 2)}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.11.2

Cancel the common factor.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 5 (8 x) \frac{2 (15 x + 2)}{5} + 35 \frac{2 (15 x + 2)}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.11.3

Rewrite the expression.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 8 x (2 (15 x + 2)) + 35 \frac{2 (15 x + 2)}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.12

Multiply $2$ by $8$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 16 x (15 x + 2) + 35 \frac{2 (15 x + 2)}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.13

Cancel the common factor of $5$ .

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Step 22.2.9.13.1

Factor $5$ out of $35$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 16 x (15 x + 2) + 5 (7) \frac{2 (15 x + 2)}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.13.2

Cancel the common factor.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 16 x (15 x + 2) + 5 \cdot 7 \frac{2 (15 x + 2)}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.13.3

Rewrite the expression.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 16 x (15 x + 2) + 7 (2 (15 x + 2)))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.14

Multiply $2$ by $7$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 16 x (15 x + 2) + 14 (15 x + 2))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.15

Simplify each term.

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Step 22.2.9.15.1

Apply the distributive property.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 16 x (15 x) + 16 x \cdot 2 + 14 (15 x + 2))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.15.2

Rewrite using the commutative property of multiplication.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 16 \cdot 15 x \cdot x + 16 x \cdot 2 + 14 (15 x + 2))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.15.3

Multiply $2$ by $16$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 16 \cdot 15 x \cdot x + 32 x + 14 (15 x + 2))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.15.4

Simplify each term.

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Step 22.2.9.15.4.1

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

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Step 22.2.9.15.4.1.1

Move $x$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 16 \cdot 15 (x \cdot x) + 32 x + 14 (15 x + 2))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.15.4.1.2

Multiply $x$ by $x$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 16 \cdot 15 x^{2} + 32 x + 14 (15 x + 2))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.15.4.2

Multiply $16$ by $15$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 240 x^{2} + 32 x + 14 (15 x + 2))}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.15.5

Apply the distributive property.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 240 x^{2} + 32 x + 14 (15 x) + 14 \cdot 2)}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.15.6

Multiply $15$ by $14$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 240 x^{2} + 32 x + 210 x + 14 \cdot 2)}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.15.7

Multiply $14$ by $2$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 240 x^{2} + 32 x + 210 x + 28)}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.9.16

Add $32 x$ and $210 x$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + 240 x^{2} + 242 x + 28)}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.10

Find the common denominator.

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Step 22.2.10.1

Write $240 x^{2}$ as a fraction with denominator $1$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + \frac{240 x^{2}}{1} + 242 x + 28)}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.10.2

Multiply $\frac{240 x^{2}}{1}$ by $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + \frac{240 x^{2}}{1} \cdot \frac{5}{5} + 242 x + 28)}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.10.3

Multiply $\frac{240 x^{2}}{1}$ by $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + \frac{240 x^{2} \cdot 5}{5} + 242 x + 28)}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.10.4

Write $242 x$ as a fraction with denominator $1$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + \frac{240 x^{2} \cdot 5}{5} + \frac{242 x}{1} + 28)}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.10.5

Multiply $\frac{242 x}{1}$ by $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + \frac{240 x^{2} \cdot 5}{5} + \frac{242 x}{1} \cdot \frac{5}{5} + 28)}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.10.6

Multiply $\frac{242 x}{1}$ by $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + \frac{240 x^{2} \cdot 5}{5} + \frac{242 x \cdot 5}{5} + 28)}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.10.7

Write $28$ as a fraction with denominator $1$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + \frac{240 x^{2} \cdot 5}{5} + \frac{242 x \cdot 5}{5} + \frac{28}{1})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.10.8

Multiply $\frac{28}{1}$ by $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + \frac{240 x^{2} \cdot 5}{5} + \frac{242 x \cdot 5}{5} + \frac{28}{1} \cdot \frac{5}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.10.9

Multiply $\frac{28}{1}$ by $\frac{5}{5}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} (\frac{4 x (15 x + 4)}{5} + \frac{240 x^{2} \cdot 5}{5} + \frac{242 x \cdot 5}{5} + \frac{28 \cdot 5}{5})}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.11

Combine the numerators over the common denominator.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{4 x (15 x + 4) + 240 x^{2} \cdot 5 + 242 x \cdot 5 + 28 \cdot 5}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.12

Simplify each term.

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Step 22.2.12.1

Apply the distributive property.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{4 x (15 x) + 4 x \cdot 4 + 240 x^{2} \cdot 5 + 242 x \cdot 5 + 28 \cdot 5}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.12.2

Rewrite using the commutative property of multiplication.

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{4 \cdot 15 x \cdot x + 4 x \cdot 4 + 240 x^{2} \cdot 5 + 242 x \cdot 5 + 28 \cdot 5}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.12.3

Multiply $4$ by $4$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{4 \cdot 15 x \cdot x + 16 x + 240 x^{2} \cdot 5 + 242 x \cdot 5 + 28 \cdot 5}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.12.4

Simplify each term.

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Step 22.2.12.4.1

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

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Step 22.2.12.4.1.1

Move $x$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{4 \cdot 15 (x \cdot x) + 16 x + 240 x^{2} \cdot 5 + 242 x \cdot 5 + 28 \cdot 5}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.12.4.1.2

Multiply $x$ by $x$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{4 \cdot 15 x^{2} + 16 x + 240 x^{2} \cdot 5 + 242 x \cdot 5 + 28 \cdot 5}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.12.4.2

Multiply $4$ by $15$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{60 x^{2} + 16 x + 240 x^{2} \cdot 5 + 242 x \cdot 5 + 28 \cdot 5}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.12.5

Multiply $5$ by $240$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{60 x^{2} + 16 x + 1200 x^{2} + 242 x \cdot 5 + 28 \cdot 5}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.12.6

Multiply $5$ by $242$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{60 x^{2} + 16 x + 1200 x^{2} + 1210 x + 28 \cdot 5}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.12.7

Multiply $28$ by $5$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{60 x^{2} + 16 x + 1200 x^{2} + 1210 x + 140}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.13

Add $60 x^{2}$ and $1200 x^{2}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{1260 x^{2} + 16 x + 1210 x + 140}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.14

Add $16 x$ and $1210 x$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{1260 x^{2} + 1226 x + 140}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.15

Factor $2$ out of $1260 x^{2} + 1226 x + 140$ .

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Step 22.2.15.1

Factor $2$ out of $1260 x^{2}$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{2 (630 x^{2}) + 1226 x + 140}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.15.2

Factor $2$ out of $1226 x$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{2 (630 x^{2}) + 2 (613 x) + 140}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.15.3

Factor $2$ out of $140$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{2 (630 x^{2}) + 2 (613 x) + 2 (70)}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.15.4

Factor $2$ out of $2 (630 x^{2}) + 2 (613 x)$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{2 (630 x^{2} + 613 x) + 2 (70)}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.15.5

Factor $2$ out of $2 (630 x^{2} + 613 x) + 2 (70)$ .

$\frac{x^{4} {(\frac{15 x + 4}{5})}^{4} \frac{2 (630 x^{2} + 613 x + 70)}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.16

Combine exponents.

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Step 22.2.16.1

Combine $\frac{2 (630 x^{2} + 613 x + 70)}{5}$ and $x^{4}$ .

$\frac{\frac{2 (630 x^{2} + 613 x + 70) x^{4}}{5} {(\frac{15 x + 4}{5})}^{4}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.16.2

Combine $\frac{2 (630 x^{2} + 613 x + 70) x^{4}}{5}$ and ${(\frac{15 x + 4}{5})}^{4}$ .

$\frac{\frac{2 (630 x^{2} + 613 x + 70) x^{4} {(\frac{15 x + 4}{5})}^{4}}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.17

Apply the product rule to $\frac{15 x + 4}{5}$ .

$\frac{\frac{2 (630 x^{2} + 613 x + 70) x^{4} \frac{{(15 x + 4)}^{4}}{5^{4}}}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.18

Combine exponents.

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Step 22.2.18.1

Combine $2$ and $\frac{{(15 x + 4)}^{4}}{5^{4}}$ .

$\frac{\frac{(630 x^{2} + 613 x + 70) x^{4} \frac{2 {(15 x + 4)}^{4}}{5^{4}}}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.18.2

Combine $\frac{2 {(15 x + 4)}^{4}}{5^{4}}$ and $x^{4}$ .

$\frac{\frac{(630 x^{2} + 613 x + 70) \frac{2 {(15 x + 4)}^{4} x^{4}}{5^{4}}}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.19

Raise $5$ to the power of $4$ .

$\frac{\frac{(630 x^{2} + 613 x + 70) \frac{2 {(15 x + 4)}^{4} x^{4}}{625}}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.20

Factor $\frac{2 {(15 x + 4)}^{4} x^{4}}{625}$ out of $\frac{(630 x^{2} + 613 x + 70) \frac{2 {(15 x + 4)}^{4} x^{4}}{625}}{5}$ .

$\frac{\frac{2 {(15 x + 4)}^{4} x^{4}}{625} \cdot \frac{630 x^{2} + 613 x + 70}{5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.21

Multiply $\frac{2 {(15 x + 4)}^{4} x^{4}}{625} \cdot \frac{630 x^{2} + 613 x + 70}{5}$ .

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Step 22.2.21.1

Multiply $\frac{2 {(15 x + 4)}^{4} x^{4}}{625}$ by $\frac{630 x^{2} + 613 x + 70}{5}$ .

$\frac{\frac{2 {(15 x + 4)}^{4} x^{4} (630 x^{2} + 613 x + 70)}{625 \cdot 5}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.2.21.2

Multiply $625$ by $5$ .

$\frac{\frac{2 {(15 x + 4)}^{4} x^{4} (630 x^{2} + 613 x + 70)}{3125}}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.3

Combine terms.

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Step 22.3.1

Rewrite $\frac{\frac{2 {(15 x + 4)}^{4} x^{4} (630 x^{2} + 613 x + 70)}{3125}}{{(8 x + 7)}^{\frac{1}{2}}}$ as a product.

$\frac{2 {(15 x + 4)}^{4} x^{4} (630 x^{2} + 613 x + 70)}{3125} \cdot \frac{1}{{(8 x + 7)}^{\frac{1}{2}}}$

Step 22.3.2

Multiply $\frac{2 {(15 x + 4)}^{4} x^{4} (630 x^{2} + 613 x + 70)}{3125}$ by $\frac{1}{{(8 x + 7)}^{\frac{1}{2}}}$ .

$\frac{2 {(15 x + 4)}^{4} x^{4} (630 x^{2} + 613 x + 70)}{3125 {(8 x + 7)}^{\frac{1}{2}}}$

Step 22.4

Reorder terms.

$\frac{2 x^{4} {(15 x + 4)}^{4} (630 x^{2} + 613 x + 70)}{3125 {(8 x + 7)}^{\frac{1}{2}}}$