Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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 $f (x) = \ln (x)$ and $g (x) = x^{2}$ .

$\frac{x^{2} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 1.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.2.1

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

$\frac{x^{2} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 1.2.2

Multiply $2$ by $2$ .

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

Step 1.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

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

Step 1.4

Differentiate using the Power Rule.

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.4.2.1

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

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

Step 1.4.2.2

Cancel the common factors.

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Step 1.4.2.2.1

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

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

Step 1.4.2.2.2

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

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

Step 1.4.2.2.3

Cancel the common factor.

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

Step 1.4.2.2.4

Rewrite the expression.

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

Step 1.4.2.2.5

Divide $x$ by $1$ .

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

Step 1.4.3

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

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

Step 1.4.4

Simplify with factoring out.

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Step 1.4.4.1

Multiply $2$ by $- 1$ .

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

Step 1.4.4.2

Factor $x$ out of $x - 2 \ln (x) x$ .

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Step 1.4.4.2.1

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

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

Step 1.4.4.2.2

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

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

Step 1.4.4.2.3

Factor $x$ out of $- 2 \ln (x) x$ .

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

Step 1.4.4.2.4

Factor $x$ out of $x \cdot 1 + x (- 2 \ln (x))$ .

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

Step 1.5

Cancel the common factors.

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Step 1.5.1

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

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

Step 1.5.2

Cancel the common factor.

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

Step 1.5.3

Rewrite the expression.

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

Step 1.6

Simplify $- 2 \ln (x)$ by moving $2$ inside the logarithm.

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

Step 2.2

Differentiate.

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Step 2.2.1

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

Multiply $3$ by $2$ .

$f'' (x) = \frac{x^{3} \frac{d}{d x} (1 - \ln (x^{2})) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.2.2

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

$f'' (x) = \frac{x^{3} (\frac{d}{d x} (1) + \frac{d}{d x} (- \ln (x^{2}))) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.2.3

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

$f'' (x) = \frac{x^{3} (0 + \frac{d}{d x} (- \ln (x^{2}))) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.2.4

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

$f'' (x) = \frac{x^{3} \frac{d}{d x} (- \ln (x^{2})) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.2.5

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

$f'' (x) = \frac{x^{3} (- \frac{d}{d x} \ln (x^{2})) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.3

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

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Step 2.3.1

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

$f'' (x) = \frac{x^{3} (- (\frac{d}{d u} (\ln (u)) \frac{d}{d x} (x^{2}))) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.3.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$f'' (x) = \frac{x^{3} (- (\frac{1}{u} \cdot \frac{d}{d x} (x^{2}))) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.3.3

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

$f'' (x) = \frac{x^{3} (- (\frac{1}{x^{2}} \cdot \frac{d}{d x} (x^{2}))) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.4

Differentiate using the Power Rule.

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Step 2.4.1

Combine $x^{3}$ and $\frac{1}{x^{2}}$ .

$f'' (x) = \frac{- \frac{x^{3}}{x^{2}} \cdot \frac{d}{d x} x^{2} - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.4.2

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 2.4.2.1

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

$f'' (x) = \frac{- \frac{x^{2} x}{x^{2}} \cdot \frac{d}{d x} x^{2} - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.4.2.2

Cancel the common factors.

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Step 2.4.2.2.1

Multiply by $1$ .

$f'' (x) = \frac{- \frac{x^{2} x}{x^{2} \cdot 1} \cdot \frac{d}{d x} x^{2} - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.4.2.2.2

Cancel the common factor.

$f'' (x) = \frac{- \frac{x^{2} x}{x^{2} \cdot 1} \cdot \frac{d}{d x} x^{2} - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.4.2.2.3

Rewrite the expression.

$f'' (x) = \frac{- \frac{x}{1} \cdot \frac{d}{d x} x^{2} - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.4.2.2.4

Divide $x$ by $1$ .

$f'' (x) = \frac{- x \frac{d}{d x} x^{2} - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.4.3

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

$f'' (x) = \frac{- x (2 x) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.4.4

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{- 2 x \cdot x - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.5

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

$f'' (x) = \frac{- 2 (x \cdot x) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.6

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

$f'' (x) = \frac{- 2 (x \cdot x) - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.7

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

$f'' (x) = \frac{- 2 x^{1 + 1} - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.8

Add $1$ and $1$ .

$f'' (x) = \frac{- 2 x^{2} - (1 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 2.9

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

$f'' (x) = \frac{- 2 x^{2} - (1 - \ln (x^{2})) (3 x^{2})}{x^{6}}$

Step 2.10

Simplify with factoring out.

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Step 2.10.1

Multiply $3$ by $- 1$ .

$f'' (x) = \frac{- 2 x^{2} - 3 (1 - \ln (x^{2})) x^{2}}{x^{6}}$

Step 2.10.2

Factor $x^{2}$ out of $- 2 x^{2} - 3 (1 - \ln (x^{2})) x^{2}$ .

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Step 2.10.2.1

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

$f'' (x) = \frac{x^{2} \cdot - 2 - 3 (1 - \ln (x^{2})) x^{2}}{x^{6}}$

Step 2.10.2.2

Factor $x^{2}$ out of $- 3 (1 - \ln (x^{2})) x^{2}$ .

$f'' (x) = \frac{x^{2} \cdot - 2 + x^{2} (- 3 (1 - \ln (x^{2})))}{x^{6}}$

Step 2.10.2.3

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

$f'' (x) = \frac{x^{2} (- 2 - 3 (1 - \ln (x^{2})))}{x^{6}}$

Step 2.11

Cancel the common factors.

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Step 2.11.1

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

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

Step 2.11.2

Cancel the common factor.

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

Step 2.11.3

Rewrite the expression.

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

Step 2.12

Simplify.

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Step 2.12.1

Apply the distributive property.

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

Step 2.12.2

Simplify the numerator.

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Step 2.12.2.1

Simplify each term.

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Step 2.12.2.1.1

Multiply $- 3$ by $1$ .

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

Step 2.12.2.1.2

Multiply $- 3 (- \ln (x^{2}))$ .

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Step 2.12.2.1.2.1

Multiply $- 1$ by $- 3$ .

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

Step 2.12.2.1.2.2

Simplify $3 \ln (x^{2})$ by moving $3$ inside the logarithm.

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

Step 2.12.2.1.3

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 2.12.2.1.3.1

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

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

Step 2.12.2.1.3.2

Multiply $2$ by $3$ .

$f'' (x) = \frac{- 2 - 3 + \ln (x^{6})}{x^{4}}$

Step 2.12.2.2

Subtract $3$ from $- 2$ .

$f'' (x) = \frac{- 5 + \ln (x^{6})}{x^{4}}$

Step 2.12.3

Rewrite $- 5$ as $- 1 (5)$ .

$f'' (x) = \frac{- 1 \cdot 5 + \ln (x^{6})}{x^{4}}$

Step 2.12.4

Factor $- 1$ out of $\ln (x^{6})$ .

$f'' (x) = \frac{- 1 \cdot 5 - 1 (- \ln (x^{6}))}{x^{4}}$

Step 2.12.5

Factor $- 1$ out of $- 1 (5) - 1 (- \ln (x^{6}))$ .

$f'' (x) = \frac{- 1 (5 - \ln (x^{6}))}{x^{4}}$

Step 2.12.6

Move the negative in front of the fraction.

$f'' (x) = - \frac{5 - \ln (x^{6})}{x^{4}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{1 - \ln (x^{2})}{x^{3}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

$\frac{x^{2} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 4.1.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 4.1.2.1

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

$\frac{x^{2} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 4.1.2.2

Multiply $2$ by $2$ .

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

Step 4.1.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

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

Step 4.1.4

Differentiate using the Power Rule.

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Step 4.1.4.1

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

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

Step 4.1.4.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 4.1.4.2.1

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

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

Step 4.1.4.2.2

Cancel the common factors.

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Step 4.1.4.2.2.1

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

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

Step 4.1.4.2.2.2

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

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

Step 4.1.4.2.2.3

Cancel the common factor.

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

Step 4.1.4.2.2.4

Rewrite the expression.

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

Step 4.1.4.2.2.5

Divide $x$ by $1$ .

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

Step 4.1.4.3

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

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

Step 4.1.4.4

Simplify with factoring out.

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Step 4.1.4.4.1

Multiply $2$ by $- 1$ .

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

Step 4.1.4.4.2

Factor $x$ out of $x - 2 \ln (x) x$ .

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Step 4.1.4.4.2.1

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

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

Step 4.1.4.4.2.2

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

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

Step 4.1.4.4.2.3

Factor $x$ out of $- 2 \ln (x) x$ .

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

Step 4.1.4.4.2.4

Factor $x$ out of $x \cdot 1 + x (- 2 \ln (x))$ .

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

Step 4.1.5

Cancel the common factors.

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Step 4.1.5.1

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

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

Step 4.1.5.2

Cancel the common factor.

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

Step 4.1.5.3

Rewrite the expression.

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

Step 4.1.6

Simplify $- 2 \ln (x)$ by moving $2$ inside the logarithm.

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1 - \ln (x^{2})}{x^{3}}$ .

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

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{1 - \ln (x^{2})}{x^{3}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{1 - \ln (x^{2})}{x^{3}} = 0$

Step 5.2

Set the numerator equal to zero.

$1 - \ln (x^{2}) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Subtract $1$ from both sides of the equation.

$- \ln (x^{2}) = - 1$

Step 5.3.2

Divide each term in $- \ln (x^{2}) = - 1$ by $- 1$ and simplify.

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Step 5.3.2.1

Divide each term in $- \ln (x^{2}) = - 1$ by $- 1$ .

$\frac{- \ln (x^{2})}{- 1} = \frac{- 1}{- 1}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{\ln (x^{2})}{1} = \frac{- 1}{- 1}$

Step 5.3.2.2.2

Divide $\ln (x^{2})$ by $1$ .

$\ln (x^{2}) = \frac{- 1}{- 1}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $- 1$ by $- 1$ .

$\ln (x^{2}) = 1$

Step 5.3.3

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x^{2})} = e^{1}$

Step 5.3.4

Rewrite $\ln (x^{2}) = 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{1} = x^{2}$

Step 5.3.5

Solve for $x$ .

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Step 5.3.5.1

Rewrite the equation as $x^{2} = e^{1}$ .

$x^{2} = e$

Step 5.3.5.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{e^{1}}$

Step 5.3.5.3

Simplify.

$x = \pm \sqrt{e}$

Step 5.3.5.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.3.5.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{e}$

Step 5.3.5.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{e}$

Step 5.3.5.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{e}, - \sqrt{e}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{1 - \ln (x^{2})}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 6.2.2

Simplify $\sqrt[3]{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 6.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 6.3

Set the argument in $\ln (x^{2})$ less than or equal to $0$ to find where the expression is undefined.

$x^{2} \leq 0$

Step 6.4

Solve for $x$ .

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Step 6.4.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{0}$

Step 6.4.2

Simplify the equation.

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Step 6.4.2.1

Simplify the left side.

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Step 6.4.2.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{0}$

Step 6.4.2.2

Simplify the right side.

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Step 6.4.2.2.1

Simplify $\sqrt{0}$ .

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Step 6.4.2.2.1.1

Rewrite $0$ as $0^{2}$ .

$| x | \leq \sqrt{0^{2}}$

Step 6.4.2.2.1.2

Pull terms out from under the radical.

$| x | \leq | 0 |$

Step 6.4.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| x | \leq 0$

Step 6.4.3

Write $| x | \leq 0$ as a piecewise.

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Step 6.4.3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 6.4.3.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 0$

Step 6.4.3.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 6.4.3.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 0$

Step 6.4.3.5

Write as a piecewise.

${\begin{cases} x \leq 0 & x \geq 0 \\ - x \leq 0 & x < 0 \end{cases}$

Step 6.4.4

Find the intersection of $x \leq 0$ and $x \geq 0$ .

$x = 0$

Step 6.4.5

Solve $- x \leq 0$ when $x < 0$ .

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Step 6.4.5.1

Divide each term in $- x \leq 0$ by $- 1$ and simplify.

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Step 6.4.5.1.1

Divide each term in $- x \leq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{0}{- 1}$

Step 6.4.5.1.2

Simplify the left side.

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Step 6.4.5.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{0}{- 1}$

Step 6.4.5.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{0}{- 1}$

Step 6.4.5.1.3

Simplify the right side.

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Step 6.4.5.1.3.1

Divide $0$ by $- 1$ .

$x \geq 0$

Step 6.4.5.2

Find the intersection of $x \geq 0$ and $x < 0$ .

No solution

Step 6.4.6

Find the union of the solutions.

$x = 0$

Step 7

Critical points to evaluate.

$x = \sqrt{e}$

Step 8

Evaluate the second derivative at $x = \sqrt{e}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{5 - \ln ({(\sqrt{e})}^{6})}{{(\sqrt{e})}^{4}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Rewrite ${\sqrt{e}}^{6}$ as $e^{3}$ .

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Step 9.1.1.1

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

$- \frac{5 - \ln ({(e^{\frac{1}{2}})}^{6})}{{\sqrt{e}}^{4}}$

Step 9.1.1.2

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

$- \frac{5 - \ln (e^{\frac{1}{2} \cdot 6})}{{\sqrt{e}}^{4}}$

Step 9.1.1.3

Combine $\frac{1}{2}$ and $6$ .

$- \frac{5 - \ln (e^{\frac{6}{2}})}{{\sqrt{e}}^{4}}$

Step 9.1.1.4

Cancel the common factor of $6$ and $2$ .

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Step 9.1.1.4.1

Factor $2$ out of $6$ .

$- \frac{5 - \ln (e^{\frac{2 \cdot 3}{2}})}{{\sqrt{e}}^{4}}$

Step 9.1.1.4.2

Cancel the common factors.

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Step 9.1.1.4.2.1

Factor $2$ out of $2$ .

$- \frac{5 - \ln (e^{\frac{2 \cdot 3}{2 (1)}})}{{\sqrt{e}}^{4}}$

Step 9.1.1.4.2.2

Cancel the common factor.

$- \frac{5 - \ln (e^{\frac{2 \cdot 3}{2 \cdot 1}})}{{\sqrt{e}}^{4}}$

Step 9.1.1.4.2.3

Rewrite the expression.

$- \frac{5 - \ln (e^{\frac{3}{1}})}{{\sqrt{e}}^{4}}$

Step 9.1.1.4.2.4

Divide $3$ by $1$ .

$- \frac{5 - \ln (e^{3})}{{\sqrt{e}}^{4}}$

Step 9.1.2

Use logarithm rules to move $3$ out of the exponent.

$- \frac{5 - (3 \ln (e))}{{\sqrt{e}}^{4}}$

Step 9.1.3

The natural logarithm of $e$ is $1$ .

$- \frac{5 - (3 \cdot 1)}{{\sqrt{e}}^{4}}$

Step 9.1.4

Multiply $3$ by $1$ .

$- \frac{5 - 1 \cdot 3}{{\sqrt{e}}^{4}}$

Step 9.1.5

Multiply $- 1$ by $3$ .

$- \frac{5 - 3}{{\sqrt{e}}^{4}}$

Step 9.1.6

Subtract $3$ from $5$ .

$- \frac{2}{{\sqrt{e}}^{4}}$

Step 9.2

Rewrite ${\sqrt{e}}^{4}$ as $e^{2}$ .

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Step 9.2.1

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

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

Step 9.2.2

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

$- \frac{2}{e^{\frac{1}{2} \cdot 4}}$

Step 9.2.3

Combine $\frac{1}{2}$ and $4$ .

$- \frac{2}{e^{\frac{4}{2}}}$

Step 9.2.4

Cancel the common factor of $4$ and $2$ .

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Step 9.2.4.1

Factor $2$ out of $4$ .

$- \frac{2}{e^{\frac{2 \cdot 2}{2}}}$

Step 9.2.4.2

Cancel the common factors.

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Step 9.2.4.2.1

Factor $2$ out of $2$ .

$- \frac{2}{e^{\frac{2 \cdot 2}{2 (1)}}}$

Step 9.2.4.2.2

Cancel the common factor.

$- \frac{2}{e^{\frac{2 \cdot 2}{2 \cdot 1}}}$

Step 9.2.4.2.3

Rewrite the expression.

$- \frac{2}{e^{\frac{2}{1}}}$

Step 9.2.4.2.4

Divide $2$ by $1$ .

$- \frac{2}{e^{2}}$

Step 10

$x = \sqrt{e}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \sqrt{e}$ is a local maximum

Step 11

Find the y-value when $x = \sqrt{e}$ .

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Step 11.1

Replace the variable $x$ with $\sqrt{e}$ in the expression.

$f (\sqrt{e}) = \frac{\ln (\sqrt{e})}{{(\sqrt{e})}^{2}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Rewrite ${\sqrt{e}}^{2}$ as $e$ .

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Step 11.2.1.1

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

$f (\sqrt{e}) = \frac{\ln (\sqrt{e})}{{(e^{\frac{1}{2}})}^{2}}$

Step 11.2.1.2

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

$f (\sqrt{e}) = \frac{\ln (\sqrt{e})}{e^{\frac{1}{2} \cdot 2}}$

Step 11.2.1.3

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

$f (\sqrt{e}) = \frac{\ln (\sqrt{e})}{e^{\frac{2}{2}}}$

Step 11.2.1.4

Cancel the common factor of $2$ .

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Step 11.2.1.4.1

Cancel the common factor.

$f (\sqrt{e}) = \frac{\ln (\sqrt{e})}{e^{\frac{2}{2}}}$

Step 11.2.1.4.2

Rewrite the expression.

$f (\sqrt{e}) = \frac{\ln (\sqrt{e})}{e}$

Step 11.2.1.5

Simplify.

$f (\sqrt{e}) = \frac{\ln (\sqrt{e})}{e}$

Step 11.2.2

The final answer is $\frac{\ln (\sqrt{e})}{e}$ .

$y = \frac{\ln (\sqrt{e})}{e}$

Step 12

These are the local extrema for $f (x) = \frac{\ln (x)}{x^{2}}$ .

$(\sqrt{e}, \frac{\ln (\sqrt{e})}{e})$ is a local maxima

Step 13