Calculus Examples

$k (x) = \frac{\ln (x)}{\sqrt{x}}$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 1.4

Simplify.

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

Step 1.5

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

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

Step 1.6

Combine fractions.

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 1.7

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 1.7.1

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

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Step 1.7.1.1

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

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

Step 1.7.1.2

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

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

Step 1.7.2

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

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

Step 1.7.3

Combine the numerators over the common denominator.

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

Step 1.7.4

Subtract $1$ from $2$ .

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

Step 1.8

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

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

Step 1.9

Combine.

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

Step 1.10

Apply the distributive property.

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

Step 1.11

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 1.11.1

Cancel the common factor.

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

Step 1.11.2

Rewrite the expression.

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

Step 1.12

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

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Step 1.12.1

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

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Step 1.12.1.1

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

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

Step 1.12.1.2

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

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

Step 1.12.2

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

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

Step 1.12.3

Combine the numerators over the common denominator.

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

Step 1.12.4

Add $1$ and $2$ .

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

Step 1.13

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

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

Step 1.14

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

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

Step 1.15

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

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

Step 1.16

Combine the numerators over the common denominator.

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

Step 1.17

Simplify the numerator.

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Step 1.17.1

Multiply $- 1$ by $2$ .

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

Step 1.17.2

Subtract $2$ from $1$ .

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

Step 1.18

Simplify terms.

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Step 1.18.1

Move the negative in front of the fraction.

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

Step 1.18.2

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

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

Step 1.18.3

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

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

Step 1.18.4

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

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

Step 1.18.5

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

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

Step 1.18.6

Cancel the common factor.

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

Step 1.18.7

Rewrite the expression.

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

Step 1.19

Simplify each term.

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Step 1.19.1

Rewrite $\frac{\ln (x)}{2}$ as $\frac{1}{2} \ln (x)$ .

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

Step 1.19.2

Simplify $\frac{1}{2} \ln (x)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.1

Cancel the common factor.

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

Step 2.2.1.2.2

Rewrite the expression.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Add $0$ and $\frac{d}{d x} [- \ln (x^{\frac{1}{2}})]$ .

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

Step 2.2.5

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

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

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $x^{\frac{3}{2}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.6.1

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

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

Step 2.6.2

Combine the numerators over the common denominator.

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

Step 2.6.3

Subtract $1$ from $3$ .

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

Step 2.6.4

Divide $2$ by $2$ .

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

Step 2.7

Simplify $x^{1}$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Combine the numerators over the common denominator.

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

Step 2.12

Simplify the numerator.

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Step 2.12.1

Multiply $- 1$ by $2$ .

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

Step 2.12.2

Subtract $2$ from $1$ .

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

Step 2.13

Move the negative in front of the fraction.

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

Multiply $x^{- \frac{1}{2}}$ by $x$ by adding the exponents.

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Step 2.16.1

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

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Step 2.16.1.1

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

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

Step 2.16.1.2

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

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

Step 2.16.2

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

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

Step 2.16.3

Combine the numerators over the common denominator.

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

Step 2.16.4

Add $- 1$ and $2$ .

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

Step 2.17

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

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

Step 2.18

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

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

Step 2.19

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

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

Step 2.20

Combine the numerators over the common denominator.

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

Step 2.21

Simplify the numerator.

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Step 2.21.1

Multiply $- 1$ by $2$ .

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

Step 2.21.2

Subtract $2$ from $3$ .

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

Step 2.22

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

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

Step 2.23

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

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

Step 2.24

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

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

Step 2.25

Combine the numerators over the common denominator.

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

Step 2.26

Multiply $2$ by $- 1$ .

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

Step 2.27

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

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

Step 2.28

Multiply $- 2$ by $3$ .

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

Step 2.29

Factor $2$ out of $- 6 x^{\frac{1}{2}}$ .

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

Step 2.30

Cancel the common factors.

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Step 2.30.1

Factor $2$ out of $2$ .

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

Step 2.30.2

Cancel the common factor.

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

Step 2.30.3

Rewrite the expression.

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

Step 2.30.4

Divide $- 3 x^{\frac{1}{2}}$ by $1$ .

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

Step 2.31

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

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

Step 2.32

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

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

Step 2.33

Simplify.

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Step 2.33.1

Apply the distributive property.

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

Step 2.33.2

Simplify the numerator.

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Step 2.33.2.1

Simplify each term.

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Step 2.33.2.1.1

Multiply $- 3$ by $1$ .

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

Step 2.33.2.1.2

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

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Step 2.33.2.1.2.1

Multiply $- 1$ by $- 3$ .

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

Step 2.33.2.1.2.2

Simplify $3 \ln (x^{\frac{1}{2}})$ by moving $3$ inside the logarithm.

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

Step 2.33.2.1.3

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

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Step 2.33.2.1.3.1

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

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

Step 2.33.2.1.3.2

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

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

Step 2.33.2.2

Subtract $3 x^{\frac{1}{2}}$ from $- x^{\frac{1}{2}}$ .

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

Step 2.33.3

Reorder terms.

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

Step 2.33.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \ln (x^{\frac{3}{2}}) - 4 x^{\frac{1}{2}}$ .

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Step 2.33.4.1

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

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

Step 2.33.4.2

Factor $x^{\frac{1}{2}}$ out of $- 4 x^{\frac{1}{2}}$ .

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

Step 2.33.4.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (\ln (x^{\frac{3}{2}})) + x^{\frac{1}{2}} \cdot - 4$ .

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

Step 2.33.5

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

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

Step 2.33.6

Multiply $x^{3}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.33.6.1

Move $x^{- \frac{1}{2}}$ .

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

Step 2.33.6.2

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

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

Step 2.33.6.3

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

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

Step 2.33.6.4

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

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

Step 2.33.6.5

Combine the numerators over the common denominator.

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

Step 2.33.6.6

Simplify the numerator.

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Step 2.33.6.6.1

Multiply $3$ by $2$ .

$k'' (x) = \frac{\ln (x^{\frac{3}{2}}) - 4}{2 x^{\frac{- 1 + 6}{2}}}$

Step 2.33.6.6.2

Add $- 1$ and $6$ .

$k'' (x) = \frac{\ln (x^{\frac{3}{2}}) - 4}{2 x^{\frac{5}{2}}}$

Step 3

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

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

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

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

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

Step 4.1.2

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

Cancel the common factor of $2$ .

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Step 4.1.3.2.1

Cancel the common factor.

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

Step 4.1.3.2.2

Rewrite the expression.

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

Step 4.1.4

Simplify.

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

Step 4.1.5

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

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

Step 4.1.6

Combine fractions.

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Step 4.1.6.1

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

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

Step 4.1.6.2

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

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

Step 4.1.7

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.1.7.1

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

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Step 4.1.7.1.1

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

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

Step 4.1.7.1.2

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

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

Step 4.1.7.2

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

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

Step 4.1.7.3

Combine the numerators over the common denominator.

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

Step 4.1.7.4

Subtract $1$ from $2$ .

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

Step 4.1.8

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

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

Step 4.1.9

Combine.

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

Step 4.1.10

Apply the distributive property.

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

Step 4.1.11

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 4.1.11.1

Cancel the common factor.

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

Step 4.1.11.2

Rewrite the expression.

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

Step 4.1.12

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

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Step 4.1.12.1

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

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Step 4.1.12.1.1

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

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

Step 4.1.12.1.2

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

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

Step 4.1.12.2

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

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

Step 4.1.12.3

Combine the numerators over the common denominator.

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

Step 4.1.12.4

Add $1$ and $2$ .

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

Step 4.1.13

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

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

Step 4.1.14

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

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

Step 4.1.15

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

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

Step 4.1.16

Combine the numerators over the common denominator.

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

Step 4.1.17

Simplify the numerator.

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Step 4.1.17.1

Multiply $- 1$ by $2$ .

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

Step 4.1.17.2

Subtract $2$ from $1$ .

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

Step 4.1.18

Simplify terms.

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Step 4.1.18.1

Move the negative in front of the fraction.

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

Step 4.1.18.2

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

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

Step 4.1.18.3

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

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

Step 4.1.18.4

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

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

Step 4.1.18.5

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

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

Step 4.1.18.6

Cancel the common factor.

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

Step 4.1.18.7

Rewrite the expression.

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

Step 4.1.19

Simplify each term.

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Step 4.1.19.1

Rewrite $\frac{\ln (x)}{2}$ as $\frac{1}{2} \ln (x)$ .

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

Step 4.1.19.2

Simplify $\frac{1}{2} \ln (x)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

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

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Subtract $1$ from both sides of the equation.

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

Step 5.3.2

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

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Step 5.3.2.1

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

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

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.

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

Step 5.3.2.2.2

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

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

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $- 1$ by $- 1$ .

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

Step 5.3.3

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

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

Step 5.3.4

Rewrite $\ln (x^{\frac{1}{2}}) = 1$ 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 $b^{y} = x$ .

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

Step 5.3.5

Solve for $x$ .

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Step 5.3.5.1

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

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

Step 5.3.5.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 5.3.5.3

Simplify the exponent.

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Step 5.3.5.3.1

Simplify the left side.

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Step 5.3.5.3.1.1

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

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Step 5.3.5.3.1.1.1

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

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Step 5.3.5.3.1.1.1.1

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

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

Step 5.3.5.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.3.5.3.1.1.1.2.1

Cancel the common factor.

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

Step 5.3.5.3.1.1.1.2.2

Rewrite the expression.

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

Step 5.3.5.3.1.1.2

Simplify.

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

Step 5.3.5.3.2

Simplify the right side.

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Step 5.3.5.3.2.1

Multiply the exponents in ${(e^{1})}^{2}$ .

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Step 5.3.5.3.2.1.1

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

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

Step 5.3.5.3.2.1.2

Multiply $2$ by $1$ .

$x = e^{2}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 6.1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 6.1.3

Anything raised to $1$ is the base itself.

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

Step 6.2

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

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

Step 6.3

Solve for $x$ .

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Step 6.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{3}}}^{2} = 0^{2}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

$x^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.2.1

Cancel the common factor.

$x^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.2.2

Rewrite the expression.

$x^{3} = 0^{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

$x^{3} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

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

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

Step 6.3.3.2

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

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Step 6.3.3.2.1

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

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

Step 6.3.3.2.2

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

$x = 0$

Step 6.4

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

$\sqrt{x} \leq 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

To remove the radical on the left side of the inequality, square both sides of the inequality.

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

Step 6.5.2

Simplify each side of the inequality.

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Step 6.5.2.1

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

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

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

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

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Step 6.5.2.2.1.1

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

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Step 6.5.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} \leq 0^{2}$

Step 6.5.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.5.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} \leq 0^{2}$

Step 6.5.2.2.1.1.2.2

Rewrite the expression.

$x^{1} \leq 0^{2}$

Step 6.5.2.2.1.2

Simplify.

$x \leq 0^{2}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

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

$x \leq 0$

Step 6.6

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 6.7

Set the radicand in $\sqrt{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 6.8

Solve for $x$ .

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Step 6.8.1

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

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

Step 6.8.2

Simplify the equation.

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Step 6.8.2.1

Simplify the left side.

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Step 6.8.2.1.1

Pull terms out from under the radical.

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

Step 6.8.2.2

Simplify the right side.

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Step 6.8.2.2.1

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

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Step 6.8.2.2.1.1

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

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

Step 6.8.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 6.9

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = e^{2}$

Step 8

Evaluate the second derivative at $x = e^{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply the exponents in ${(e^{2})}^{\frac{3}{2}}$ .

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Step 9.1.1.1

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

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

Step 9.1.1.2

Cancel the common factor of $2$ .

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Step 9.1.1.2.1

Cancel the common factor.

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

Step 9.1.1.2.2

Rewrite the expression.

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

Step 9.1.2

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

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

Step 9.1.3

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

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

Step 9.1.4

Multiply $3$ by $1$ .

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

Step 9.1.5

Subtract $4$ from $3$ .

$\frac{- 1}{2 {(e^{2})}^{\frac{5}{2}}}$

Step 9.2

Simplify the expression.

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Step 9.2.1

Multiply the exponents in ${(e^{2})}^{\frac{5}{2}}$ .

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Step 9.2.1.1

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

$\frac{- 1}{2 e^{2 (\frac{5}{2})}}$

Step 9.2.1.2

Cancel the common factor of $2$ .

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Step 9.2.1.2.1

Cancel the common factor.

$\frac{- 1}{2 e^{2 (\frac{5}{2})}}$

Step 9.2.1.2.2

Rewrite the expression.

$\frac{- 1}{2 e^{5}}$

Step 9.2.2

Move the negative in front of the fraction.

$- \frac{1}{2 e^{5}}$

Step 10

$x = e^{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = e^{2}$ is a local maximum

Step 11

Find the y-value when $x = e^{2}$ .

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Step 11.1

Replace the variable $x$ with $e^{2}$ in the expression.

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

Step 11.2

Simplify the result.

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Step 11.2.1

Remove parentheses.

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

Step 11.2.2

Simplify the numerator.

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Step 11.2.2.1

Use logarithm rules to move $2$ out of the exponent.

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

Step 11.2.2.2

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

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

Step 11.2.2.3

Multiply $2$ by $1$ .

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

Step 11.2.3

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

$k (e^{2}) = \frac{2}{e}$

Step 11.2.4

The final answer is $\frac{2}{e}$ .

$y = \frac{2}{e}$

Step 12

These are the local extrema for $k (x) = \frac{\ln (x)}{\sqrt{x}}$ .

$(e^{2}, \frac{2}{e})$ is a local maxima

Step 13