Calculus Examples

${(3 - 2 x^{2})}^{2} \sqrt{2 x^{2} + 1}$

Step 1

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

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

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 $f (x) = {(3 - 2 x^{2})}^{2}$ and $g (x) = {(2 x^{2} + 1)}^{\frac{1}{2}}$ .

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

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 $g (x) = 2 x^{2} + 1$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $2 x^{2} + 1$ .

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

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 $n = \frac{1}{2}$ .

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

Step 3.3

Replace all occurrences of $u_{1}$ with $2 x^{2} + 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Multiply $2$ by $2$ .

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

Step 13

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

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

Step 14

Simplify terms.

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Step 14.1

Add $4 x$ and $0$ .

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

Step 14.2

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

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

Step 14.3

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

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

Step 14.4

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

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

Step 15

Cancel the common factors.

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Step 15.1

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

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

Step 15.2

Cancel the common factor.

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

Step 15.3

Rewrite the expression.

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

Step 16

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

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Step 16.1

To apply the Chain Rule, set $u_{2}$ as $3 - 2 x^{2}$ .

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

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 $n = 2$ .

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

Step 16.3

Replace all occurrences of $u_{2}$ with $3 - 2 x^{2}$ .

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

Step 17

Differentiate.

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Step 17.1

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

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

Step 17.2

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

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

Step 17.3

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

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

Step 17.4

Add $0$ and $\frac{d}{d x} [- 2 x^{2}]$ .

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

Step 17.5

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

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

Step 17.6

Multiply $- 2$ by $2$ .

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

Step 17.7

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

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

Step 17.8

Multiply $2$ by $- 4$ .

$\frac{2 {(3 - 2 x^{2})}^{2} x}{{(2 x^{2} + 1)}^{\frac{1}{2}}} - 8 {(2 x^{2} + 1)}^{\frac{1}{2}} (3 - 2 x^{2}) x$

Step 18

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

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

Step 19

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

$\frac{2 {(3 - 2 x^{2})}^{2} x}{{(2 x^{2} + 1)}^{\frac{1}{2}}} + \frac{- 8 {(2 x^{2} + 1)}^{\frac{1}{2}} (3 - 2 x^{2}) x {(2 x^{2} + 1)}^{\frac{1}{2}}}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 20

Combine the numerators over the common denominator.

$\frac{2 {(3 - 2 x^{2})}^{2} x - 8 {(2 x^{2} + 1)}^{\frac{1}{2}} (3 - 2 x^{2}) x {(2 x^{2} + 1)}^{\frac{1}{2}}}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 21

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

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Step 21.1

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

$\frac{2 {(3 - 2 x^{2})}^{2} x - 8 ({(2 x^{2} + 1)}^{\frac{1}{2}} {(2 x^{2} + 1)}^{\frac{1}{2}}) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 21.2

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

$\frac{2 {(3 - 2 x^{2})}^{2} x - 8 {(2 x^{2} + 1)}^{\frac{1}{2} + \frac{1}{2}} (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 21.3

Combine the numerators over the common denominator.

$\frac{2 {(3 - 2 x^{2})}^{2} x - 8 {(2 x^{2} + 1)}^{\frac{1 + 1}{2}} (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 21.4

Add $1$ and $1$ .

$\frac{2 {(3 - 2 x^{2})}^{2} x - 8 {(2 x^{2} + 1)}^{\frac{2}{2}} (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 21.5

Divide $2$ by $2$ .

$\frac{2 {(3 - 2 x^{2})}^{2} x - 8 {(2 x^{2} + 1)}^{1} (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 22

Simplify $- 8 {(2 x^{2} + 1)}^{1} (3 - 2 x^{2}) x$ .

$\frac{2 {(3 - 2 x^{2})}^{2} x - 8 (2 x^{2} + 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23

Simplify.

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Step 23.1

Apply the distributive property.

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

Step 23.2

Simplify the numerator.

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Step 23.2.1

Simplify each term.

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Step 23.2.1.1

Rewrite ${(3 - 2 x^{2})}^{2}$ as $(3 - 2 x^{2}) (3 - 2 x^{2})$ .

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

Step 23.2.1.2

Expand $(3 - 2 x^{2}) (3 - 2 x^{2})$ using the FOIL Method.

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Step 23.2.1.2.1

Apply the distributive property.

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

Step 23.2.1.2.2

Apply the distributive property.

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

Step 23.2.1.2.3

Apply the distributive property.

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

Step 23.2.1.3

Simplify and combine like terms.

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Step 23.2.1.3.1

Simplify each term.

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Step 23.2.1.3.1.1

Multiply $3$ by $3$ .

$\frac{2 (9 + 3 (- 2 x^{2}) - 2 x^{2} \cdot 3 - 2 x^{2} (- 2 x^{2})) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.3.1.2

Multiply $- 2$ by $3$ .

$\frac{2 (9 - 6 x^{2} - 2 x^{2} \cdot 3 - 2 x^{2} (- 2 x^{2})) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.3.1.3

Multiply $3$ by $- 2$ .

$\frac{2 (9 - 6 x^{2} - 6 x^{2} - 2 x^{2} (- 2 x^{2})) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{2 (9 - 6 x^{2} - 6 x^{2} - 2 \cdot - 2 x^{2} x^{2}) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.3.1.5

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

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Step 23.2.1.3.1.5.1

Move $x^{2}$ .

$\frac{2 (9 - 6 x^{2} - 6 x^{2} - 2 \cdot - 2 (x^{2} x^{2})) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.3.1.5.2

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

$\frac{2 (9 - 6 x^{2} - 6 x^{2} - 2 \cdot - 2 x^{2 + 2}) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.3.1.5.3

Add $2$ and $2$ .

$\frac{2 (9 - 6 x^{2} - 6 x^{2} - 2 \cdot - 2 x^{4}) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.3.1.6

Multiply $- 2$ by $- 2$ .

$\frac{2 (9 - 6 x^{2} - 6 x^{2} + 4 x^{4}) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.3.2

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

$\frac{2 (9 - 12 x^{2} + 4 x^{4}) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.4

Apply the distributive property.

$\frac{(2 \cdot 9 + 2 (- 12 x^{2}) + 2 (4 x^{4})) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.5

Simplify.

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Step 23.2.1.5.1

Multiply $2$ by $9$ .

$\frac{(18 + 2 (- 12 x^{2}) + 2 (4 x^{4})) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.5.2

Multiply $- 12$ by $2$ .

$\frac{(18 - 24 x^{2} + 2 (4 x^{4})) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.5.3

Multiply $4$ by $2$ .

$\frac{(18 - 24 x^{2} + 8 x^{4}) x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.6

Apply the distributive property.

$\frac{18 x - 24 x^{2} x + 8 x^{4} x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.7

Simplify.

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Step 23.2.1.7.1

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

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Step 23.2.1.7.1.1

Move $x$ .

$\frac{18 x - 24 (x \cdot x^{2}) + 8 x^{4} x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.7.1.2

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

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Step 23.2.1.7.1.2.1

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

$\frac{18 x - 24 (x^{1} x^{2}) + 8 x^{4} x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.7.1.2.2

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

$\frac{18 x - 24 x^{1 + 2} + 8 x^{4} x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.7.1.3

Add $1$ and $2$ .

$\frac{18 x - 24 x^{3} + 8 x^{4} x + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.7.2

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

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Step 23.2.1.7.2.1

Move $x$ .

$\frac{18 x - 24 x^{3} + 8 (x \cdot x^{4}) + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.7.2.2

Multiply $x$ by $x^{4}$ .

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Step 23.2.1.7.2.2.1

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

$\frac{18 x - 24 x^{3} + 8 (x^{1} x^{4}) + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.7.2.2.2

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

$\frac{18 x - 24 x^{3} + 8 x^{1 + 4} + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.7.2.3

Add $1$ and $4$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 8 (2 x^{2}) - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.8

Simplify each term.

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Step 23.2.1.8.1

Multiply $2$ by $- 8$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 16 x^{2} - 8 \cdot 1) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.8.2

Multiply $- 8$ by $1$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 16 x^{2} - 8) (3 - 2 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.9

Expand $(- 16 x^{2} - 8) (3 - 2 x^{2})$ using the FOIL Method.

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Step 23.2.1.9.1

Apply the distributive property.

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 16 x^{2} (3 - 2 x^{2}) - 8 (3 - 2 x^{2})) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.9.2

Apply the distributive property.

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 16 x^{2} \cdot 3 - 16 x^{2} (- 2 x^{2}) - 8 (3 - 2 x^{2})) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.9.3

Apply the distributive property.

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 16 x^{2} \cdot 3 - 16 x^{2} (- 2 x^{2}) - 8 \cdot 3 - 8 (- 2 x^{2})) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.10

Simplify and combine like terms.

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Step 23.2.1.10.1

Simplify each term.

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Step 23.2.1.10.1.1

Multiply $3$ by $- 16$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 48 x^{2} - 16 x^{2} (- 2 x^{2}) - 8 \cdot 3 - 8 (- 2 x^{2})) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.10.1.2

Rewrite using the commutative property of multiplication.

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 48 x^{2} - 16 \cdot - 2 x^{2} x^{2} - 8 \cdot 3 - 8 (- 2 x^{2})) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.10.1.3

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

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Step 23.2.1.10.1.3.1

Move $x^{2}$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 48 x^{2} - 16 \cdot - 2 (x^{2} x^{2}) - 8 \cdot 3 - 8 (- 2 x^{2})) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.10.1.3.2

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

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 48 x^{2} - 16 \cdot - 2 x^{2 + 2} - 8 \cdot 3 - 8 (- 2 x^{2})) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.10.1.3.3

Add $2$ and $2$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 48 x^{2} - 16 \cdot - 2 x^{4} - 8 \cdot 3 - 8 (- 2 x^{2})) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.10.1.4

Multiply $- 16$ by $- 2$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 48 x^{2} + 32 x^{4} - 8 \cdot 3 - 8 (- 2 x^{2})) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.10.1.5

Multiply $- 8$ by $3$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 48 x^{2} + 32 x^{4} - 24 - 8 (- 2 x^{2})) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.10.1.6

Multiply $- 2$ by $- 8$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 48 x^{2} + 32 x^{4} - 24 + 16 x^{2}) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.10.2

Add $- 48 x^{2}$ and $16 x^{2}$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} + (- 32 x^{2} + 32 x^{4} - 24) x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.11

Apply the distributive property.

$\frac{18 x - 24 x^{3} + 8 x^{5} - 32 x^{2} x + 32 x^{4} x - 24 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.12

Simplify.

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Step 23.2.1.12.1

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

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Step 23.2.1.12.1.1

Move $x$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} - 32 (x \cdot x^{2}) + 32 x^{4} x - 24 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.12.1.2

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

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Step 23.2.1.12.1.2.1

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

$\frac{18 x - 24 x^{3} + 8 x^{5} - 32 (x^{1} x^{2}) + 32 x^{4} x - 24 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.12.1.2.2

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

$\frac{18 x - 24 x^{3} + 8 x^{5} - 32 x^{1 + 2} + 32 x^{4} x - 24 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.12.1.3

Add $1$ and $2$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} - 32 x^{3} + 32 x^{4} x - 24 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.12.2

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

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Step 23.2.1.12.2.1

Move $x$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} - 32 x^{3} + 32 (x \cdot x^{4}) - 24 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.12.2.2

Multiply $x$ by $x^{4}$ .

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Step 23.2.1.12.2.2.1

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

$\frac{18 x - 24 x^{3} + 8 x^{5} - 32 x^{3} + 32 (x^{1} x^{4}) - 24 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.12.2.2.2

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

$\frac{18 x - 24 x^{3} + 8 x^{5} - 32 x^{3} + 32 x^{1 + 4} - 24 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.1.12.2.3

Add $1$ and $4$ .

$\frac{18 x - 24 x^{3} + 8 x^{5} - 32 x^{3} + 32 x^{5} - 24 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.2

Subtract $24 x$ from $18 x$ .

$\frac{- 24 x^{3} + 8 x^{5} - 32 x^{3} + 32 x^{5} - 6 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.3

Subtract $32 x^{3}$ from $- 24 x^{3}$ .

$\frac{8 x^{5} - 56 x^{3} + 32 x^{5} - 6 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.2.4

Add $8 x^{5}$ and $32 x^{5}$ .

$\frac{40 x^{5} - 56 x^{3} - 6 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3

Simplify the numerator.

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Step 23.3.1

Factor $2 x$ out of $40 x^{5} - 56 x^{3} - 6 x$ .

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Step 23.3.1.1

Factor $2 x$ out of $40 x^{5}$ .

$\frac{2 x (20 x^{4}) - 56 x^{3} - 6 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.1.2

Factor $2 x$ out of $- 56 x^{3}$ .

$\frac{2 x (20 x^{4}) + 2 x (- 28 x^{2}) - 6 x}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.1.3

Factor $2 x$ out of $- 6 x$ .

$\frac{2 x (20 x^{4}) + 2 x (- 28 x^{2}) + 2 x (- 3)}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.1.4

Factor $2 x$ out of $2 x (20 x^{4}) + 2 x (- 28 x^{2})$ .

$\frac{2 x (20 x^{4} - 28 x^{2}) + 2 x (- 3)}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.1.5

Factor $2 x$ out of $2 x (20 x^{4} - 28 x^{2}) + 2 x (- 3)$ .

$\frac{2 x (20 x^{4} - 28 x^{2} - 3)}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{2 x (20 {(x^{2})}^{2} - 28 x^{2} - 3)}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$\frac{2 x (20 u^{2} - 28 u - 3)}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.4

Factor by grouping.

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Step 23.3.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 20 \cdot - 3 = - 60$ and whose sum is $b = - 28$ .

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Step 23.3.4.1.1

Factor $- 28$ out of $- 28 u$ .

$\frac{2 x (20 u^{2} - 28 (u) - 3)}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.4.1.2

Rewrite $- 28$ as $2$ plus $- 30$

$\frac{2 x (20 u^{2} + (2 - 30) u - 3)}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.4.1.3

Apply the distributive property.

$\frac{2 x (20 u^{2} + 2 u - 30 u - 3)}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.4.2

Factor out the greatest common factor from each group.

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Step 23.3.4.2.1

Group the first two terms and the last two terms.

$\frac{2 x ((20 u^{2} + 2 u) - 30 u - 3)}{{(2 x^{2} + 1)}^{\frac{1}{2}}}$

Step 23.3.4.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 23.3.4.3

Factor the polynomial by factoring out the greatest common factor, $10 u + 1$ .

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

Step 23.3.5

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

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