Calculus Examples

$y = \frac{2 x}{\sqrt{x^{2} + 2}}$

Step 1

Write $y = \frac{2 x}{\sqrt{x^{2} + 2}}$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

Cancel the common factor of $2$ .

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Step 2.1.3.2.1

Cancel the common factor.

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

Step 2.1.3.2.2

Rewrite the expression.

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

Step 2.1.4

Simplify.

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

Step 2.1.5

Differentiate using the Power Rule.

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.6

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

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.7

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

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

Step 2.1.8

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

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

Step 2.1.9

Combine the numerators over the common denominator.

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

Step 2.1.10

Simplify the numerator.

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Step 2.1.10.1

Multiply $- 1$ by $2$ .

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

Step 2.1.10.2

Subtract $2$ from $1$ .

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

Step 2.1.11

Combine fractions.

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Step 2.1.11.1

Move the negative in front of the fraction.

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

Step 2.1.11.2

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

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

Step 2.1.11.3

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

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

Step 2.1.11.4

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

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

Step 2.1.12

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

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

Step 2.1.13

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

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

Step 2.1.14

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f' (x) = 2 (\frac{{(x^{2} + 2)}^{\frac{1}{2}} - \frac{x}{2 {(x^{2} + 2)}^{\frac{1}{2}}} \cdot (2 x + 0)}{x^{2} + 2})$

Step 2.1.15

Combine fractions.

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Step 2.1.15.1

Add $2 x$ and $0$ .

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

Step 2.1.15.2

Multiply $2$ by $- 1$ .

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

Step 2.1.15.3

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

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

Step 2.1.15.4

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

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

Step 2.1.16

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

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

Step 2.1.17

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

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

Step 2.1.18

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

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

Step 2.1.19

Add $1$ and $1$ .

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

Step 2.1.20

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

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

Step 2.1.21

Cancel the common factors.

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Step 2.1.21.1

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

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

Step 2.1.21.2

Cancel the common factor.

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

Step 2.1.21.3

Rewrite the expression.

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

Step 2.1.22

Move the negative in front of the fraction.

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

Step 2.1.23

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

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

Step 2.1.24

Combine the numerators over the common denominator.

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

Step 2.1.25

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

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Step 2.1.25.1

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

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

Step 2.1.25.2

Combine the numerators over the common denominator.

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

Step 2.1.25.3

Add $1$ and $1$ .

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

Step 2.1.25.4

Divide $2$ by $2$ .

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

Step 2.1.26

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

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

Step 2.1.27

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

$f' (x) = 2 (\frac{\frac{0 + 2}{{(x^{2} + 2)}^{\frac{1}{2}}}}{x^{2} + 2})$

Step 2.1.28

Add $0$ and $2$ .

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

Step 2.1.29

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

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

Step 2.1.30

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

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

Step 2.1.31

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

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Step 2.1.31.1

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

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Step 2.1.31.1.1

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

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

Step 2.1.31.1.2

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

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

Step 2.1.31.2

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

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

Step 2.1.31.3

Combine the numerators over the common denominator.

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

Step 2.1.31.4

Add $1$ and $2$ .

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

Step 2.1.32

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

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

Step 2.1.33

Multiply $2$ by $2$ .

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

Step 2.2

Find the second derivative.

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Step 2.2.1

Differentiate using the Constant Multiple Rule.

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Step 2.2.1.1

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

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

Step 2.2.1.2

Apply basic rules of exponents.

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Step 2.2.1.2.1

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

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

Step 2.2.1.2.2

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

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Step 2.2.1.2.2.1

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

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

Step 2.2.1.2.2.2

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

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Step 2.2.1.2.2.2.1

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

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

Step 2.2.1.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 2.2.1.2.2.3

Move the negative in front of the fraction.

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Combine the numerators over the common denominator.

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

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.2.6.2

Subtract $2$ from $- 3$ .

$f'' (x) = 4 (- \frac{3}{2} \cdot {(x^{2} + 2)}^{\frac{- 5}{2}} \frac{d}{d x} (x^{2} + 2))$

Step 2.2.7

Move the negative in front of the fraction.

$f'' (x) = 4 (- \frac{3}{2} \cdot {(x^{2} + 2)}^{- \frac{5}{2}} \frac{d}{d x} (x^{2} + 2))$

Step 2.2.8

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

$f'' (x) = 4 (- \frac{{(x^{2} + 2)}^{- \frac{5}{2}} \cdot 3}{2} \cdot \frac{d}{d x} (x^{2} + 2))$

Step 2.2.9

Simplify the expression.

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Step 2.2.9.1

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

$f'' (x) = 4 (- \frac{3 \cdot {(x^{2} + 2)}^{- \frac{5}{2}}}{2} \cdot \frac{d}{d x} (x^{2} + 2))$

Step 2.2.9.2

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

$f'' (x) = 4 (- \frac{3}{2 {(x^{2} + 2)}^{\frac{5}{2}}} \cdot \frac{d}{d x} (x^{2} + 2))$

Step 2.2.9.3

Multiply $- 1$ by $4$ .

$f'' (x) = - 4 (\frac{3}{2 {(x^{2} + 2)}^{\frac{5}{2}}} \cdot \frac{d}{d x} (x^{2} + 2))$

Step 2.2.10

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

$f'' (x) = \frac{3 \cdot - 4}{2 {(x^{2} + 2)}^{\frac{5}{2}}} \cdot \frac{d}{d x} (x^{2} + 2)$

Step 2.2.11

Multiply $3$ by $- 4$ .

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

Step 2.2.12

Factor $2$ out of $- 12$ .

$f'' (x) = \frac{2 \cdot - 6}{2 {(x^{2} + 2)}^{\frac{5}{2}}} \cdot \frac{d}{d x} (x^{2} + 2)$

Step 2.2.13

Cancel the common factors.

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Step 2.2.13.1

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

$f'' (x) = \frac{2 \cdot - 6}{2 ({(x^{2} + 2)}^{\frac{5}{2}})} \cdot \frac{d}{d x} (x^{2} + 2)$

Step 2.2.13.2

Cancel the common factor.

$f'' (x) = \frac{2 \cdot - 6}{2 {(x^{2} + 2)}^{\frac{5}{2}}} \cdot \frac{d}{d x} (x^{2} + 2)$

Step 2.2.13.3

Rewrite the expression.

$f'' (x) = \frac{- 6}{{(x^{2} + 2)}^{\frac{5}{2}}} \cdot \frac{d}{d x} (x^{2} + 2)$

Step 2.2.14

Move the negative in front of the fraction.

$f'' (x) = - \frac{6}{{(x^{2} + 2)}^{\frac{5}{2}}} \cdot \frac{d}{d x} (x^{2} + 2)$

Step 2.2.15

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

$f'' (x) = - \frac{6}{{(x^{2} + 2)}^{\frac{5}{2}}} \cdot (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (2))$

Step 2.2.16

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

$f'' (x) = - \frac{6}{{(x^{2} + 2)}^{\frac{5}{2}}} \cdot (2 x + \frac{d}{d x} (2))$

Step 2.2.17

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{6}{{(x^{2} + 2)}^{\frac{5}{2}}} \cdot (2 x + 0)$

Step 2.2.18

Combine fractions.

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Step 2.2.18.1

Add $2 x$ and $0$ .

$f'' (x) = - \frac{6}{{(x^{2} + 2)}^{\frac{5}{2}}} \cdot (2 x)$

Step 2.2.18.2

Multiply $2$ by $- 1$ .

$f'' (x) = - 2 \frac{6}{{(x^{2} + 2)}^{\frac{5}{2}}} \cdot x$

Step 2.2.18.3

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

$f'' (x) = \frac{- 2 \cdot 6}{{(x^{2} + 2)}^{\frac{5}{2}}} \cdot x$

Step 2.2.18.4

Multiply $- 2$ by $6$ .

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

Step 2.2.18.5

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

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

Step 2.2.18.6

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 3

Set the second derivative equal to $0$ then solve the equation $- \frac{12 x}{{(x^{2} + 2)}^{\frac{5}{2}}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- \frac{12 x}{{(x^{2} + 2)}^{\frac{5}{2}}} = 0$

Step 3.2

Set the numerator equal to zero.

$12 x = 0$

Step 3.3

Divide each term in $12 x = 0$ by $12$ and simplify.

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Step 3.3.1

Divide each term in $12 x = 0$ by $12$ .

$\frac{12 x}{12} = \frac{0}{12}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $12$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{0}{12}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{12}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $12$ .

$x = 0$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $0$ in $f (x) = \frac{2 x}{\sqrt{x^{2} + 2}}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{2 (0)}{\sqrt{{(0)}^{2} + 2}}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $2$ by $0$ .

$f (0) = \frac{0}{\sqrt{0^{2} + 2}}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = \frac{0}{\sqrt{0 + 2}}$

Step 4.1.2.2.2

Add $0$ and $2$ .

$f (0) = \frac{0}{\sqrt{2}}$

Step 4.1.2.3

Divide $0$ by $\sqrt{2}$ .

$f (0) = 0$

Step 4.1.2.4

The final answer is $0$ .

$0$

Step 4.2

The point found by substituting $0$ in $f (x) = \frac{2 x}{\sqrt{x^{2} + 2}}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.1$ in the expression.

$f'' (- 0.1) = - \frac{12 (- 0.1)}{{({(- 0.1)}^{2} + 2)}^{\frac{5}{2}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $12$ by $- 0.1$ .

$f'' (- 0.1) = - \frac{- 1.2}{{({(- 0.1)}^{2} + 2)}^{\frac{5}{2}}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Raise $- 0.1$ to the power of $2$ .

$f'' (- 0.1) = - \frac{- 1.2}{{(0.01 + 2)}^{\frac{5}{2}}}$

Step 6.2.2.2

Add $0.01$ and $2$ .

$f'' (- 0.1) = - \frac{- 1.2}{{2.01}^{\frac{5}{2}}}$

Step 6.2.2.3

Rewrite $2.01$ as ${1.41774468}^{2}$ .

$f'' (- 0.1) = - \frac{- 1.2}{{({1.41774468}^{2})}^{\frac{5}{2}}}$

Step 6.2.2.4

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

$f'' (- 0.1) = - \frac{- 1.2}{{1.41774468}^{2 (\frac{5}{2})}}$

Step 6.2.2.5

Cancel the common factor of $2$ .

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Step 6.2.2.5.1

Cancel the common factor.

$f'' (- 0.1) = - \frac{- 1.2}{{1.41774468}^{2 (\frac{5}{2})}}$

Step 6.2.2.5.2

Rewrite the expression.

$f'' (- 0.1) = - \frac{- 1.2}{{1.41774468}^{5}}$

Step 6.2.2.6

Raise $1.41774468$ to the power of $5$ .

$f'' (- 0.1) = - \frac{- 1.2}{5.72783031}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Divide $- 1.2$ by $5.72783031$ .

$f'' (- 0.1) = 0.20950341$

Step 6.2.3.2

Multiply $- 1$ by $- 0.20950341$ .

$f'' (- 0.1) = 0.20950341$

Step 6.2.4

The final answer is $0.20950341$ .

$0.20950341$

Step 6.3

At $- 0.1$ , the second derivative is $0.20950341$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 0)$ .

Increasing on $(- \infty, 0)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = - \frac{12 (0.1)}{{({(0.1)}^{2} + 2)}^{\frac{5}{2}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $12$ by $0.1$ .

$f'' (0.1) = - \frac{1.2}{{({0.1}^{2} + 2)}^{\frac{5}{2}}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Raise $0.1$ to the power of $2$ .

$f'' (0.1) = - \frac{1.2}{{(0.01 + 2)}^{\frac{5}{2}}}$

Step 7.2.2.2

Add $0.01$ and $2$ .

$f'' (0.1) = - \frac{1.2}{{2.01}^{\frac{5}{2}}}$

Step 7.2.2.3

Rewrite $2.01$ as ${1.41774468}^{2}$ .

$f'' (0.1) = - \frac{1.2}{{({1.41774468}^{2})}^{\frac{5}{2}}}$

Step 7.2.2.4

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

$f'' (0.1) = - \frac{1.2}{{1.41774468}^{2 (\frac{5}{2})}}$

Step 7.2.2.5

Cancel the common factor of $2$ .

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Step 7.2.2.5.1

Cancel the common factor.

$f'' (0.1) = - \frac{1.2}{{1.41774468}^{2 (\frac{5}{2})}}$

Step 7.2.2.5.2

Rewrite the expression.

$f'' (0.1) = - \frac{1.2}{{1.41774468}^{5}}$

Step 7.2.2.6

Raise $1.41774468$ to the power of $5$ .

$f'' (0.1) = - \frac{1.2}{5.72783031}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Divide $1.2$ by $5.72783031$ .

$f'' (0.1) = - 1 \cdot 0.20950341$

Step 7.2.3.2

Multiply $- 1$ by $0.20950341$ .

$f'' (0.1) = - 0.20950341$

Step 7.2.4

The final answer is $- 0.20950341$ .

$- 0.20950341$

Step 7.3

At $0.1$ , the second derivative is $- 0.20950341$ . Since this is negative, the second derivative is decreasing on the interval $(0, \infty)$

Decreasing on $(0, \infty)$ since $f'' (x) < 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(0, 0)$ .

$(0, 0)$

Step 9