Calculus Examples

$f (x) = \frac{\ln (x)}{11 x}$

Step 1

Find the first derivative.

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Step 1.1

Since $\frac{1}{11}$ is constant with respect to $x$ , the derivative of $\frac{\ln (x)}{11 x}$ with respect to $x$ is $\frac{1}{11} \frac{d}{d x} [\frac{\ln (x)}{x}]$ .

$f' (x) = \frac{1}{11} \cdot \frac{d}{d x} (\frac{\ln (x)}{x})$

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 $g (x) = x$ .

$f' (x) = \frac{1}{11} \cdot \frac{x \frac{d}{d x} (\ln (x)) - \ln (x) \frac{d}{d x} x}{x^{2}}$

Step 1.3

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

$f' (x) = \frac{1}{11} \cdot \frac{x (\frac{1}{x}) - \ln (x) \frac{d}{d x} x}{x^{2}}$

Step 1.4

Differentiate using the Power Rule.

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Step 1.4.1

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

$f' (x) = \frac{1}{11} \cdot \frac{\frac{x}{x} - \ln (x) \frac{d}{d x} x}{x^{2}}$

Step 1.4.2

Cancel the common factor of $x$ .

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Step 1.4.2.1

Cancel the common factor.

$f' (x) = \frac{1}{11} \cdot \frac{\frac{x}{x} - \ln (x) \frac{d}{d x} x}{x^{2}}$

Step 1.4.2.2

Rewrite the expression.

$f' (x) = \frac{1}{11} \cdot \frac{1 - \ln (x) \frac{d}{d x} x}{x^{2}}$

Step 1.4.3

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

$f' (x) = \frac{1}{11} \cdot \frac{1 - \ln (x) \cdot 1}{x^{2}}$

Step 1.4.4

Combine fractions.

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Step 1.4.4.1

Multiply $- 1$ by $1$ .

$f' (x) = \frac{1}{11} \cdot \frac{1 - \ln (x)}{x^{2}}$

Step 1.4.4.2

Multiply $\frac{1}{11}$ by $\frac{1 - \ln (x)}{x^{2}}$ .

$f' (x) = \frac{1 - \ln (x)}{11 x^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{1}{11}$ is constant with respect to $x$ , the derivative of $\frac{1 - \ln (x)}{11 x^{2}}$ with respect to $x$ is $\frac{1}{11} \frac{d}{d x} [\frac{1 - \ln (x)}{x^{2}}]$ .

$f'' (x) = \frac{1}{11} \cdot \frac{d}{d x} (\frac{1 - \ln (x)}{x^{2}})$

Step 2.2

$f'' (x) = \frac{1}{11} \cdot \frac{x^{2} \frac{d}{d x} (1 - \ln (x)) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{{(x^{2})}^{2}}$

Step 2.3

Differentiate.

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Step 2.3.1

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

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Step 2.3.1.1

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

$f'' (x) = \frac{1}{11} \cdot \frac{x^{2} \frac{d}{d x} (1 - \ln (x)) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{2 \cdot 2}}$

Step 2.3.1.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{1}{11} \cdot \frac{x^{2} \frac{d}{d x} (1 - \ln (x)) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.3.2

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

$f'' (x) = \frac{1}{11} \cdot \frac{x^{2} (\frac{d}{d x} (1) + \frac{d}{d x} (- \ln (x))) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.3.3

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

$f'' (x) = \frac{1}{11} \cdot \frac{x^{2} (0 + \frac{d}{d x} (- \ln (x))) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.3.4

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

$f'' (x) = \frac{1}{11} \cdot \frac{x^{2} \frac{d}{d x} (- \ln (x)) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.3.5

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

$f'' (x) = \frac{1}{11} \cdot \frac{x^{2} (- \frac{d}{d x} \ln (x)) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4

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

$f'' (x) = \frac{1}{11} \cdot \frac{x^{2} (- \frac{1}{x}) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.5

Differentiate using the Power Rule.

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Step 2.5.1

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

$f'' (x) = \frac{1}{11} \cdot \frac{- \frac{x^{2}}{x} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.5.2

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

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Step 2.5.2.1

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

$f'' (x) = \frac{1}{11} \cdot \frac{- \frac{x \cdot x}{x} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.5.2.2

Cancel the common factors.

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Step 2.5.2.2.1

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

$f'' (x) = \frac{1}{11} \cdot \frac{- \frac{x \cdot x}{x} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.5.2.2.2

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

$f'' (x) = \frac{1}{11} \cdot \frac{- \frac{x \cdot x}{x \cdot 1} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.5.2.2.3

Cancel the common factor.

$f'' (x) = \frac{1}{11} \cdot \frac{- \frac{x \cdot x}{x \cdot 1} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.5.2.2.4

Rewrite the expression.

$f'' (x) = \frac{1}{11} \cdot \frac{- \frac{x}{1} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.5.2.2.5

Divide $x$ by $1$ .

$f'' (x) = \frac{1}{11} \cdot \frac{- x - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.5.3

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

$f'' (x) = \frac{1}{11} \cdot \frac{- x - (1 - \ln (x)) (2 x)}{x^{4}}$

Step 2.5.4

Simplify with factoring out.

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Step 2.5.4.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{1}{11} \cdot \frac{- x - 2 (1 - \ln (x)) x}{x^{4}}$

Step 2.5.4.2

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

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Step 2.5.4.2.1

Factor $x$ out of $- x$ .

$f'' (x) = \frac{1}{11} \cdot \frac{x \cdot - 1 - 2 (1 - \ln (x)) x}{x^{4}}$

Step 2.5.4.2.2

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

$f'' (x) = \frac{1}{11} \cdot \frac{x \cdot - 1 + x (- 2 (1 - \ln (x)))}{x^{4}}$

Step 2.5.4.2.3

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

$f'' (x) = \frac{1}{11} \cdot \frac{x (- 1 - 2 (1 - \ln (x)))}{x^{4}}$

Step 2.6

Cancel the common factors.

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Step 2.6.1

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

$f'' (x) = \frac{1}{11} \cdot \frac{x (- 1 - 2 (1 - \ln (x)))}{x \cdot x^{3}}$

Step 2.6.2

Cancel the common factor.

$f'' (x) = \frac{1}{11} \cdot \frac{x (- 1 - 2 (1 - \ln (x)))}{x \cdot x^{3}}$

Step 2.6.3

Rewrite the expression.

$f'' (x) = \frac{1}{11} \cdot \frac{- 1 - 2 (1 - \ln (x))}{x^{3}}$

Step 2.7

Multiply $\frac{1}{11}$ by $\frac{- 1 - 2 (1 - \ln (x))}{x^{3}}$ .

$f'' (x) = \frac{- 1 - 2 (1 - \ln (x))}{11 x^{3}}$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$f'' (x) = \frac{- 1 - 2 \cdot 1 - 2 (- \ln (x))}{11 x^{3}}$

Step 2.8.2

Simplify the numerator.

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Step 2.8.2.1

Simplify each term.

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Step 2.8.2.1.1

Multiply $- 2$ by $1$ .

$f'' (x) = \frac{- 1 - 2 - 2 (- \ln (x))}{11 x^{3}}$

Step 2.8.2.1.2

Multiply $- 2 (- \ln (x))$ .

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Step 2.8.2.1.2.1

Multiply $- 1$ by $- 2$ .

$f'' (x) = \frac{- 1 - 2 + 2 \ln (x)}{11 x^{3}}$

Step 2.8.2.1.2.2

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

$f'' (x) = \frac{- 1 - 2 + \ln (x^{2})}{11 x^{3}}$

Step 2.8.2.2

Subtract $2$ from $- 1$ .

$f'' (x) = \frac{- 3 + \ln (x^{2})}{11 x^{3}}$

Step 2.8.3

Rewrite $- 3$ as $- 1 (3)$ .

$f'' (x) = \frac{- 1 \cdot 3 + \ln (x^{2})}{11 x^{3}}$

Step 2.8.4

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

$f'' (x) = \frac{- 1 \cdot 3 - 1 (- \ln (x^{2}))}{11 x^{3}}$

Step 2.8.5

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

$f'' (x) = \frac{- 1 (3 - \ln (x^{2}))}{11 x^{3}}$

Step 2.8.6

Move the negative in front of the fraction.

$f'' (x) = - \frac{3 - \ln (x^{2})}{11 x^{3}}$

Step 3

Find the third derivative.

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Step 3.1

Since $- \frac{1}{11}$ is constant with respect to $x$ , the derivative of $- \frac{3 - \ln (x^{2})}{11 x^{3}}$ with respect to $x$ is $- \frac{1}{11} \frac{d}{d x} [\frac{3 - \ln (x^{2})}{x^{3}}]$ .

$f''' (x) = - \frac{1}{11} \cdot \frac{d}{d x} \frac{3 - \ln (x^{2})}{x^{3}}$

Step 3.2

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{3} \frac{d}{d x} (3 - \ln (x^{2})) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{{(x^{3})}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

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

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Step 3.3.1.1

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{3} \frac{d}{d x} (3 - \ln (x^{2})) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{3 \cdot 2}}$

Step 3.3.1.2

Multiply $3$ by $2$ .

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{3} \frac{d}{d x} (3 - \ln (x^{2})) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.3.2

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{3} (\frac{d}{d x} (3) + \frac{d}{d x} (- \ln (x^{2}))) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.3.3

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{3} (0 + \frac{d}{d x} (- \ln (x^{2}))) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.3.4

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{3} \frac{d}{d x} (- \ln (x^{2})) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.3.5

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{3} (- \frac{d}{d x} \ln (x^{2})) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.4

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 3.4.1

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{3} (- (\frac{d}{d u} (\ln (u)) \frac{d}{d x} (x^{2}))) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.4.2

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{3} (- (\frac{1}{u} \cdot \frac{d}{d x} (x^{2}))) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.4.3

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{3} (- (\frac{1}{x^{2}} \cdot \frac{d}{d x} (x^{2}))) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.5

Differentiate using the Power Rule.

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Step 3.5.1

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

$f''' (x) = - \frac{1}{11} \cdot \frac{- \frac{x^{3}}{x^{2}} \cdot \frac{d}{d x} x^{2} - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.5.2

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

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Step 3.5.2.1

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

$f''' (x) = - \frac{1}{11} \cdot \frac{- \frac{x^{2} x}{x^{2}} \cdot \frac{d}{d x} x^{2} - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.5.2.2

Cancel the common factors.

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Step 3.5.2.2.1

Multiply by $1$ .

$f''' (x) = - \frac{1}{11} \cdot \frac{- \frac{x^{2} x}{x^{2} \cdot 1} \cdot \frac{d}{d x} x^{2} - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.5.2.2.2

Cancel the common factor.

$f''' (x) = - \frac{1}{11} \cdot \frac{- \frac{x^{2} x}{x^{2} \cdot 1} \cdot \frac{d}{d x} x^{2} - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.5.2.2.3

Rewrite the expression.

$f''' (x) = - \frac{1}{11} \cdot \frac{- \frac{x}{1} \cdot \frac{d}{d x} x^{2} - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.5.2.2.4

Divide $x$ by $1$ .

$f''' (x) = - \frac{1}{11} \cdot \frac{- x \frac{d}{d x} x^{2} - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.5.3

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

$f''' (x) = - \frac{1}{11} \cdot \frac{- x (2 x) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.5.4

Multiply $2$ by $- 1$ .

$f''' (x) = - \frac{1}{11} \cdot \frac{- 2 x \cdot x - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.6

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

$f''' (x) = - \frac{1}{11} \cdot \frac{- 2 (x \cdot x) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.7

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

$f''' (x) = - \frac{1}{11} \cdot \frac{- 2 (x \cdot x) - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.8

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

$f''' (x) = - \frac{1}{11} \cdot \frac{- 2 x^{1 + 1} - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.9

Add $1$ and $1$ .

$f''' (x) = - \frac{1}{11} \cdot \frac{- 2 x^{2} - (3 - \ln (x^{2})) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.10

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

$f''' (x) = - \frac{1}{11} \cdot \frac{- 2 x^{2} - (3 - \ln (x^{2})) (3 x^{2})}{x^{6}}$

Step 3.11

Simplify with factoring out.

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Step 3.11.1

Multiply $3$ by $- 1$ .

$f''' (x) = - \frac{1}{11} \cdot \frac{- 2 x^{2} - 3 (3 - \ln (x^{2})) x^{2}}{x^{6}}$

Step 3.11.2

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

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Step 3.11.2.1

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{2} \cdot - 2 - 3 (3 - \ln (x^{2})) x^{2}}{x^{6}}$

Step 3.11.2.2

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{2} \cdot - 2 + x^{2} (- 3 (3 - \ln (x^{2})))}{x^{6}}$

Step 3.11.2.3

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{2} (- 2 - 3 (3 - \ln (x^{2})))}{x^{6}}$

Step 3.12

Cancel the common factors.

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Step 3.12.1

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

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{2} (- 2 - 3 (3 - \ln (x^{2})))}{x^{2} x^{4}}$

Step 3.12.2

Cancel the common factor.

$f''' (x) = - \frac{1}{11} \cdot \frac{x^{2} (- 2 - 3 (3 - \ln (x^{2})))}{x^{2} x^{4}}$

Step 3.12.3

Rewrite the expression.

$f''' (x) = - \frac{1}{11} \cdot \frac{- 2 - 3 (3 - \ln (x^{2}))}{x^{4}}$

Step 3.13

Multiply $\frac{- 2 - 3 (3 - \ln (x^{2}))}{x^{4}}$ by $\frac{1}{11}$ .

$f''' (x) = - \frac{- 2 - 3 (3 - \ln (x^{2}))}{x^{4} \cdot 11}$

Step 3.14

Move $11$ to the left of $x^{4}$ .

$f''' (x) = - \frac{- 2 - 3 (3 - \ln (x^{2}))}{11 \cdot x^{4}}$

Step 3.15

Simplify.

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Step 3.15.1

Apply the distributive property.

$f''' (x) = - \frac{- 2 - 3 \cdot 3 - 3 (- \ln (x^{2}))}{11 x^{4}}$

Step 3.15.2

Simplify the numerator.

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Step 3.15.2.1

Simplify each term.

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Step 3.15.2.1.1

Multiply $- 3$ by $3$ .

$f''' (x) = - \frac{- 2 - 9 - 3 (- \ln (x^{2}))}{11 x^{4}}$

Step 3.15.2.1.2

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

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Step 3.15.2.1.2.1

Multiply $- 1$ by $- 3$ .

$f''' (x) = - \frac{- 2 - 9 + 3 \ln (x^{2})}{11 x^{4}}$

Step 3.15.2.1.2.2

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

$f''' (x) = - \frac{- 2 - 9 + \ln ({(x^{2})}^{3})}{11 x^{4}}$

Step 3.15.2.1.3

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

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Step 3.15.2.1.3.1

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

$f''' (x) = - \frac{- 2 - 9 + \ln (x^{2 \cdot 3})}{11 x^{4}}$

Step 3.15.2.1.3.2

Multiply $2$ by $3$ .

$f''' (x) = - \frac{- 2 - 9 + \ln (x^{6})}{11 x^{4}}$

Step 3.15.2.2

Subtract $9$ from $- 2$ .

$f''' (x) = - \frac{- 11 + \ln (x^{6})}{11 x^{4}}$

Step 3.15.3

Rewrite $- 11$ as $- 1 (11)$ .

$f''' (x) = - \frac{- 1 \cdot 11 + \ln (x^{6})}{11 x^{4}}$

Step 3.15.4

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

$f''' (x) = - \frac{- 1 \cdot 11 - 1 (- \ln (x^{6}))}{11 x^{4}}$

Step 3.15.5

Factor $- 1$ out of $- 1 (11) - 1 (- \ln (x^{6}))$ .

$f''' (x) = - \frac{- 1 (11 - \ln (x^{6}))}{11 x^{4}}$

Step 3.15.6

Move the negative in front of the fraction.

$f''' (x) = \frac{11 - \ln (x^{6})}{11 x^{4}}$

Step 3.15.7

Multiply $- 1$ by $- 1$ .

$f''' (x) = 1 (\frac{11 - \ln (x^{6})}{11 x^{4}})$

Step 3.15.8

Multiply $\frac{11 - \ln (x^{6})}{11 x^{4}}$ by $1$ .

$f''' (x) = \frac{11 - \ln (x^{6})}{11 x^{4}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $\frac{1}{11}$ is constant with respect to $x$ , the derivative of $\frac{11 - \ln (x^{6})}{11 x^{4}}$ with respect to $x$ is $\frac{1}{11} \frac{d}{d x} [\frac{11 - \ln (x^{6})}{x^{4}}]$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{d}{d x} (\frac{11 - \ln (x^{6})}{x^{4}})$

Step 4.2

$f^{4} (x) = \frac{1}{11} \cdot \frac{x^{4} \frac{d}{d x} (11 - \ln (x^{6})) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{{(x^{4})}^{2}}$

Step 4.3

Differentiate.

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Step 4.3.1

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 4.3.1.1

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

$f^{4} (x) = \frac{1}{11} \cdot \frac{x^{4} \frac{d}{d x} (11 - \ln (x^{6})) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{4 \cdot 2}}$

Step 4.3.1.2

Multiply $4$ by $2$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{x^{4} \frac{d}{d x} (11 - \ln (x^{6})) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.3.2

By the Sum Rule, the derivative of $11 - \ln (x^{6})$ with respect to $x$ is $\frac{d}{d x} [11] + \frac{d}{d x} [- \ln (x^{6})]$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{x^{4} (\frac{d}{d x} (11) + \frac{d}{d x} (- \ln (x^{6}))) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.3.3

Since $11$ is constant with respect to $x$ , the derivative of $11$ with respect to $x$ is $0$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{x^{4} (0 + \frac{d}{d x} (- \ln (x^{6}))) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.3.4

Add $0$ and $\frac{d}{d x} [- \ln (x^{6})]$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{x^{4} \frac{d}{d x} (- \ln (x^{6})) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (x^{6})$ with respect to $x$ is $- \frac{d}{d x} [\ln (x^{6})]$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{x^{4} (- \frac{d}{d x} \ln (x^{6})) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.4

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 $g (x) = x^{6}$ .

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Step 4.4.1

To apply the Chain Rule, set $u$ as $x^{6}$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{x^{4} (- (\frac{d}{d u} (\ln (u)) \frac{d}{d x} (x^{6}))) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.4.2

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

$f^{4} (x) = \frac{1}{11} \cdot \frac{x^{4} (- (\frac{1}{u} \cdot \frac{d}{d x} (x^{6}))) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.4.3

Replace all occurrences of $u$ with $x^{6}$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{x^{4} (- (\frac{1}{x^{6}} \cdot \frac{d}{d x} (x^{6}))) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5

Differentiate using the Power Rule.

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Step 4.5.1

Combine $x^{4}$ and $\frac{1}{x^{6}}$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{- \frac{x^{4}}{x^{6}} \cdot \frac{d}{d x} x^{6} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.2

Cancel the common factor of $x^{4}$ and $x^{6}$ .

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Step 4.5.2.1

Multiply by $1$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{- \frac{x^{4} \cdot 1}{x^{6}} \cdot \frac{d}{d x} x^{6} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.2.2

Cancel the common factors.

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Step 4.5.2.2.1

Factor $x^{4}$ out of $x^{6}$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{- \frac{x^{4} \cdot 1}{x^{4} x^{2}} \cdot \frac{d}{d x} x^{6} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.2.2.2

Cancel the common factor.

$f^{4} (x) = \frac{1}{11} \cdot \frac{- \frac{x^{4} \cdot 1}{x^{4} x^{2}} \cdot \frac{d}{d x} x^{6} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.2.2.3

Rewrite the expression.

$f^{4} (x) = \frac{1}{11} \cdot \frac{- \frac{1}{x^{2}} \cdot \frac{d}{d x} x^{6} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{- \frac{1}{x^{2}} \cdot (6 x^{5}) - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.4

Simplify terms.

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Step 4.5.4.1

Multiply $6$ by $- 1$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{- 6 \frac{1}{x^{2}} \cdot x^{5} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.4.2

Combine $- 6$ and $\frac{1}{x^{2}}$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{\frac{- 6}{x^{2}} \cdot x^{5} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.4.3

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

$f^{4} (x) = \frac{1}{11} \cdot \frac{\frac{- 6 x^{5}}{x^{2}} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.4.4

Cancel the common factor of $x^{5}$ and $x^{2}$ .

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Step 4.5.4.4.1

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

$f^{4} (x) = \frac{1}{11} \cdot \frac{\frac{x^{2} (- 6 x^{3})}{x^{2}} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.4.4.2

Cancel the common factors.

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Step 4.5.4.4.2.1

Multiply by $1$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{\frac{x^{2} (- 6 x^{3})}{x^{2} \cdot 1} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.4.4.2.2

Cancel the common factor.

$f^{4} (x) = \frac{1}{11} \cdot \frac{\frac{x^{2} (- 6 x^{3})}{x^{2} \cdot 1} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.4.4.2.3

Rewrite the expression.

$f^{4} (x) = \frac{1}{11} \cdot \frac{\frac{- 6 x^{3}}{1} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.4.4.2.4

Divide $- 6 x^{3}$ by $1$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{- 6 x^{3} - (11 - \ln (x^{6})) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{- 6 x^{3} - (11 - \ln (x^{6})) (4 x^{3})}{x^{8}}$

Step 4.5.6

Combine fractions.

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Step 4.5.6.1

Multiply $4$ by $- 1$ .

$f^{4} (x) = \frac{1}{11} \cdot \frac{- 6 x^{3} - 4 (11 - \ln (x^{6})) x^{3}}{x^{8}}$

Step 4.5.6.2

Multiply $\frac{1}{11}$ by $\frac{- 6 x^{3} - 4 (11 - \ln (x^{6})) x^{3}}{x^{8}}$ .

$f^{4} (x) = \frac{- 6 x^{3} - 4 (11 - \ln (x^{6})) x^{3}}{11 x^{8}}$

Step 4.6

Simplify.

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Step 4.6.1

Apply the distributive property.

$f^{4} (x) = \frac{- 6 x^{3} + (- 4 \cdot 11 - 4 (- \ln (x^{6}))) x^{3}}{11 x^{8}}$

Step 4.6.2

Apply the distributive property.

$f^{4} (x) = \frac{- 6 x^{3} - 4 \cdot (11 x^{3}) - 4 (- \ln (x^{6})) x^{3}}{11 x^{8}}$

Step 4.6.3

Simplify the numerator.

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Step 4.6.3.1

Simplify each term.

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Step 4.6.3.1.1

Multiply $- 4$ by $11$ .

$f^{4} (x) = \frac{- 6 x^{3} - 44 x^{3} - 4 (- \ln (x^{6})) x^{3}}{11 x^{8}}$

Step 4.6.3.1.2

Multiply $- 4 (- \ln (x^{6}))$ .

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Step 4.6.3.1.2.1

Multiply $- 1$ by $- 4$ .

$f^{4} (x) = \frac{- 6 x^{3} - 44 x^{3} + 4 \ln (x^{6}) x^{3}}{11 x^{8}}$

Step 4.6.3.1.2.2

Simplify $4 \ln (x^{6})$ by moving $4$ inside the logarithm.

$f^{4} (x) = \frac{- 6 x^{3} - 44 x^{3} + \ln ({(x^{6})}^{4}) x^{3}}{11 x^{8}}$

Step 4.6.3.1.3

Multiply the exponents in ${(x^{6})}^{4}$ .

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Step 4.6.3.1.3.1

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

$f^{4} (x) = \frac{- 6 x^{3} - 44 x^{3} + \ln (x^{6 \cdot 4}) x^{3}}{11 x^{8}}$

Step 4.6.3.1.3.2

Multiply $6$ by $4$ .

$f^{4} (x) = \frac{- 6 x^{3} - 44 x^{3} + \ln (x^{24}) x^{3}}{11 x^{8}}$

Step 4.6.3.2

Subtract $44 x^{3}$ from $- 6 x^{3}$ .

$f^{4} (x) = \frac{- 50 x^{3} + \ln (x^{24}) x^{3}}{11 x^{8}}$

Step 4.6.3.3

Reorder factors in $- 50 x^{3} + \ln (x^{24}) x^{3}$ .

$f^{4} (x) = \frac{- 50 x^{3} + x^{3} \ln (x^{24})}{11 x^{8}}$

Step 4.6.4

Factor $x^{3}$ out of $- 50 x^{3} + x^{3} \ln (x^{24})$ .

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Step 4.6.4.1

Factor $x^{3}$ out of $- 50 x^{3}$ .

$f^{4} (x) = \frac{x^{3} \cdot - 50 + x^{3} \ln (x^{24})}{11 x^{8}}$

Step 4.6.4.2

Factor $x^{3}$ out of $x^{3} \ln (x^{24})$ .

$f^{4} (x) = \frac{x^{3} \cdot - 50 + x^{3} (\ln (x^{24}))}{11 x^{8}}$

Step 4.6.4.3

Factor $x^{3}$ out of $x^{3} \cdot - 50 + x^{3} (\ln (x^{24}))$ .

$f^{4} (x) = \frac{x^{3} (- 50 + \ln (x^{24}))}{11 x^{8}}$

Step 4.6.5

Expand $\ln (x^{24})$ by moving $24$ outside the logarithm.

$f^{4} (x) = \frac{x^{3} (- 50 + 24 \ln (x))}{11 x^{8}}$

Step 4.6.6

Cancel the common factors.

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Step 4.6.6.1

Factor $x^{3}$ out of $11 x^{8}$ .

$f^{4} (x) = \frac{x^{3} (- 50 + 24 \ln (x))}{x^{3} (11 x^{5})}$

Step 4.6.6.2

Cancel the common factor.

$f^{4} (x) = \frac{x^{3} (- 50 + 24 \ln (x))}{x^{3} (11 x^{5})}$

Step 4.6.6.3

Rewrite the expression.

$f^{4} (x) = \frac{- 50 + 24 \ln (x)}{11 x^{5}}$

Step 4.6.7

Factor $2$ out of $- 50 + 24 \ln (x)$ .

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Step 4.6.7.1

Factor $2$ out of $- 50$ .

$f^{4} (x) = \frac{2 (- 25) + 24 \ln (x)}{11 x^{5}}$

Step 4.6.7.2

Factor $2$ out of $24 \ln (x)$ .

$f^{4} (x) = \frac{2 (- 25) + 2 (12 \ln (x))}{11 x^{5}}$

Step 4.6.7.3

Factor $2$ out of $2 (- 25) + 2 (12 \ln (x))$ .

$f^{4} (x) = \frac{2 (- 25 + 12 \ln (x))}{11 x^{5}}$

Step 4.6.8

Rewrite $- 25$ as $- 1 (25)$ .

$f^{4} (x) = \frac{2 (- 1 \cdot 25 + 12 \ln (x))}{11 x^{5}}$

Step 4.6.9

Factor $- 1$ out of $12 \ln (x)$ .

$f^{4} (x) = \frac{2 (- 1 \cdot 25 - (- 12 \ln (x)))}{11 x^{5}}$

Step 4.6.10

Factor $- 1$ out of $- 1 (25) - (- 12 \ln (x))$ .

$f^{4} (x) = \frac{2 (- 1 (25 - 12 \ln (x)))}{11 x^{5}}$

Step 4.6.11

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{2 (25 - 12 \ln (x))}{11 x^{5}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $- \frac{2 (25 - 12 \ln (x))}{11 x^{5}}$ .

$- \frac{2 (25 - 12 \ln (x))}{11 x^{5}}$