Calculus Examples

$f (x) = 4 \sqrt{1 + x^{2}}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

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

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

Step 1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 {(1 + x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [{(1 + x^{2})}^{\frac{1}{2}}]$ .

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

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.6.2

Subtract $2$ from $1$ .

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

Step 1.7

Move the negative in front of the fraction.

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

Factor $2$ out of $4$ .

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

Step 1.12

Cancel the common factors.

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Step 1.12.1

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

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

Step 1.12.2

Cancel the common factor.

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

Step 1.12.3

Rewrite the expression.

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

Step 1.13

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

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

Step 1.14

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

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

Step 1.15

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

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

Step 1.16

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

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

Step 1.17

Combine fractions.

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Step 1.17.1

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

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

Step 1.17.2

Multiply $2$ by $2$ .

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

Step 1.17.3

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

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

Step 1.17.4

Reorder terms.

$f' (x) = \frac{4 x}{{(x^{2} + 1)}^{\frac{1}{2}}}$

Step 2

Find the second derivative.

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 x}{{(x^{2} + 1)}^{\frac{1}{2}}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x}{{(x^{2} + 1)}^{\frac{1}{2}}}]$ .

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

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

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

Step 2.3.2.2

Rewrite the expression.

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

Step 2.4

Simplify.

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

Step 2.5

Differentiate using the Power Rule.

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Step 2.5.1

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

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

Step 2.5.2

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

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

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

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

Step 2.10.2

Subtract $2$ from $1$ .

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

Step 2.11

Combine fractions.

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Step 2.11.1

Move the negative in front of the fraction.

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

Step 2.11.2

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

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

Step 2.11.3

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

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

Step 2.11.4

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

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

$4 \frac{{(x^{2} + 1)}^{\frac{1}{2}} - \frac{x}{2 {(x^{2} + 1)}^{\frac{1}{2}}} (2 x + 0)}{x^{2} + 1}$

Step 2.15

Combine fractions.

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Step 2.15.1

Add $2 x$ and $0$ .

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

Step 2.15.2

Multiply $2$ by $- 1$ .

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

Step 2.15.3

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

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

Step 2.15.4

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

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

Step 2.16

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

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

Step 2.17

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

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

Step 2.18

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

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

Step 2.19

Add $1$ and $1$ .

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

Step 2.20

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

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

Step 2.21

Cancel the common factors.

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Step 2.21.1

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

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

Step 2.21.2

Cancel the common factor.

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

Step 2.21.3

Rewrite the expression.

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

Step 2.22

Move the negative in front of the fraction.

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

Step 2.23

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

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

Step 2.24

Combine the numerators over the common denominator.

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

Step 2.25

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

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Step 2.25.1

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

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

Step 2.25.2

Combine the numerators over the common denominator.

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

Step 2.25.3

Add $1$ and $1$ .

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

Step 2.25.4

Divide $2$ by $2$ .

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

Step 2.26

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

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

Step 2.27

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

$4 \frac{\frac{0 + 1}{{(x^{2} + 1)}^{\frac{1}{2}}}}{x^{2} + 1}$

Step 2.28

Add $0$ and $1$ .

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

Step 2.29

Rewrite $\frac{\frac{1}{{(x^{2} + 1)}^{\frac{1}{2}}}}{x^{2} + 1}$ as a product.

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

Step 2.30

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

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

Step 2.31

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

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Step 2.31.1

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

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Step 2.31.1.1

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

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

Step 2.31.1.2

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

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

Step 2.31.2

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

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

Step 2.31.3

Combine the numerators over the common denominator.

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

Step 2.31.4

Add $1$ and $2$ .

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

Step 2.32

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

$f'' (x) = \frac{4}{{(x^{2} + 1)}^{\frac{3}{2}}}$

Step 3

Find the third derivative.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{{(x^{2} + 1)}^{\frac{3}{2}}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{{(x^{2} + 1)}^{\frac{3}{2}}}]$ .

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

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

Rewrite $\frac{1}{{(x^{2} + 1)}^{\frac{3}{2}}}$ as ${({(x^{2} + 1)}^{\frac{3}{2}})}^{- 1}$ .

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

Step 3.1.2.2

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

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Step 3.1.2.2.1

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

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

Step 3.1.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 3.1.2.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 3.1.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.1.2.2.3

Move the negative in front of the fraction.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

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

Step 3.6.2

Subtract $2$ from $- 3$ .

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

Step 3.7

Move the negative in front of the fraction.

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

Step 3.8

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

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

Step 3.9

Simplify the expression.

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Step 3.9.1

Move $3$ to the left of ${(x^{2} + 1)}^{- \frac{5}{2}}$ .

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

Step 3.9.2

Move ${(x^{2} + 1)}^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.9.3

Multiply $- 1$ by $4$ .

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

Step 3.10

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

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

Step 3.11

Multiply $3$ by $- 4$ .

$\frac{- 12}{2 {(x^{2} + 1)}^{\frac{5}{2}}} \frac{d}{d x} [x^{2} + 1]$

Step 3.12

Factor $2$ out of $- 12$ .

$\frac{2 \cdot - 6}{2 {(x^{2} + 1)}^{\frac{5}{2}}} \frac{d}{d x} [x^{2} + 1]$

Step 3.13

Cancel the common factors.

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Step 3.13.1

Factor $2$ out of $2 {(x^{2} + 1)}^{\frac{5}{2}}$ .

$\frac{2 \cdot - 6}{2 ({(x^{2} + 1)}^{\frac{5}{2}})} \frac{d}{d x} [x^{2} + 1]$

Step 3.13.2

Cancel the common factor.

$\frac{2 \cdot - 6}{2 {(x^{2} + 1)}^{\frac{5}{2}}} \frac{d}{d x} [x^{2} + 1]$

Step 3.13.3

Rewrite the expression.

$\frac{- 6}{{(x^{2} + 1)}^{\frac{5}{2}}} \frac{d}{d x} [x^{2} + 1]$

Step 3.14

Move the negative in front of the fraction.

$- \frac{6}{{(x^{2} + 1)}^{\frac{5}{2}}} \frac{d}{d x} [x^{2} + 1]$

Step 3.15

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

$- \frac{6}{{(x^{2} + 1)}^{\frac{5}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])$

Step 3.16

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

$- \frac{6}{{(x^{2} + 1)}^{\frac{5}{2}}} (2 x + \frac{d}{d x} [1])$

Step 3.17

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

$- \frac{6}{{(x^{2} + 1)}^{\frac{5}{2}}} (2 x + 0)$

Step 3.18

Combine fractions.

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Step 3.18.1

Add $2 x$ and $0$ .

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

Step 3.18.2

Multiply $2$ by $- 1$ .

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

Step 3.18.3

Combine $- 2$ and $\frac{6}{{(x^{2} + 1)}^{\frac{5}{2}}}$ .

$\frac{- 2 \cdot 6}{{(x^{2} + 1)}^{\frac{5}{2}}} x$

Step 3.18.4

Multiply $- 2$ by $6$ .

$\frac{- 12}{{(x^{2} + 1)}^{\frac{5}{2}}} x$

Step 3.18.5

Combine $\frac{- 12}{{(x^{2} + 1)}^{\frac{5}{2}}}$ and $x$ .

$\frac{- 12 x}{{(x^{2} + 1)}^{\frac{5}{2}}}$

Step 3.18.6

Move the negative in front of the fraction.

$f''' (x) = - \frac{12 x}{{(x^{2} + 1)}^{\frac{5}{2}}}$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $- \frac{12 x}{{(x^{2} + 1)}^{\frac{5}{2}}}$ .

$- \frac{12 x}{{(x^{2} + 1)}^{\frac{5}{2}}}$