Calculus Examples

$y = {(x^{3} - 12 x)}^{\ln (x)}$

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite ${(x^{3} - 12 x)}^{\ln (x)}$ as $e^{\ln ({(x^{3} - 12 x)}^{\ln (x)})}$ .

$\frac{d}{d x} [e^{\ln ({(x^{3} - 12 x)}^{\ln (x)})}]$

Step 1.2

Expand $\ln ({(x^{3} - 12 x)}^{\ln (x)})$ by moving $\ln (x)$ outside the logarithm.

$\frac{d}{d x} [e^{\ln (x) \ln (x^{3} - 12 x)}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = \ln (x) \ln (x^{3} - 12 x)$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\ln (x) \ln (x^{3} - 12 x)$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [\ln (x) \ln (x^{3} - 12 x)]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [\ln (x) \ln (x^{3} - 12 x)]$

Step 2.3

Replace all occurrences of $u_{1}$ with $\ln (x) \ln (x^{3} - 12 x)$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} \frac{d}{d x} [\ln (x) \ln (x^{3} - 12 x)]$

Step 3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \ln (x)$ and $g (x) = \ln (x^{3} - 12 x)$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\ln (x) \frac{d}{d x} [\ln (x^{3} - 12 x)] + \ln (x^{3} - 12 x) \frac{d}{d x} [\ln (x)])$

Step 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 $g (x) = x^{3} - 12 x$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $x^{3} - 12 x$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\ln (x) (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [x^{3} - 12 x]) + \ln (x^{3} - 12 x) \frac{d}{d x} [\ln (x)])$

Step 4.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\ln (x) (\frac{1}{u_{2}} \frac{d}{d x} [x^{3} - 12 x]) + \ln (x^{3} - 12 x) \frac{d}{d x} [\ln (x)])$

Step 4.3

Replace all occurrences of $u_{2}$ with $x^{3} - 12 x$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\ln (x) (\frac{1}{x^{3} - 12 x} \frac{d}{d x} [x^{3} - 12 x]) + \ln (x^{3} - 12 x) \frac{d}{d x} [\ln (x)])$

Step 5

Differentiate.

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Step 5.1

Combine $\frac{1}{x^{3} - 12 x}$ and $\ln (x)$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x)}{x^{3} - 12 x} \frac{d}{d x} [x^{3} - 12 x] + \ln (x^{3} - 12 x) \frac{d}{d x} [\ln (x)])$

Step 5.2

By the Sum Rule, the derivative of $x^{3} - 12 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x]$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x)}{x^{3} - 12 x} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x]) + \ln (x^{3} - 12 x) \frac{d}{d x} [\ln (x)])$

Step 5.3

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

$e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} + \frac{d}{d x} [- 12 x]) + \ln (x^{3} - 12 x) \frac{d}{d x} [\ln (x)])$

Step 5.4

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} - 12 \frac{d}{d x} [x]) + \ln (x^{3} - 12 x) \frac{d}{d x} [\ln (x)])$

Step 5.5

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

$e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} - 12 \cdot 1) + \ln (x^{3} - 12 x) \frac{d}{d x} [\ln (x)])$

Step 5.6

Multiply $- 12$ by $1$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} - 12) + \ln (x^{3} - 12 x) \frac{d}{d x} [\ln (x)])$

Step 6

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

$e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} - 12) + \ln (x^{3} - 12 x) \frac{1}{x})$

Step 7

Combine $\ln (x^{3} - 12 x)$ and $\frac{1}{x}$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} - 12) + \frac{\ln (x^{3} - 12 x)}{x})$

Step 8

To write $\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} - 12)$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} - 12) \cdot \frac{x}{x} + \frac{\ln (x^{3} - 12 x)}{x})$

Step 9

Combine $\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} - 12)$ and $\frac{x}{x}$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} - 12) x}{x} + \frac{\ln (x^{3} - 12 x)}{x})$

Step 10

Combine the numerators over the common denominator.

$e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\frac{\ln (x)}{x^{3} - 12 x} (3 x^{2} - 12) x + \ln (x^{3} - 12 x)}{x}$

Step 11

Combine $x$ and $\frac{\ln (x)}{x^{3} - 12 x}$ .

$e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\frac{x \ln (x)}{x^{3} - 12 x} (3 x^{2} - 12) + \ln (x^{3} - 12 x)}{x}$

Step 12

Combine $e^{\ln (x) \ln (x^{3} - 12 x)}$ and $\frac{\frac{x \ln (x)}{x^{3} - 12 x} (3 x^{2} - 12) + \ln (x^{3} - 12 x)}{x}$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{x \ln (x)}{x^{3} - 12 x} (3 x^{2} - 12) + \ln (x^{3} - 12 x))}{x}$

Step 13

Simplify.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Simplify each term.

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Step 13.1.1.1

Factor $x$ out of $x^{3} - 12 x$ .

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Step 13.1.1.1.1

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

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{x \ln (x)}{x \cdot x^{2} - 12 x} (3 x^{2} - 12) + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.1.2

Factor $x$ out of $- 12 x$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{x \ln (x)}{x \cdot x^{2} + x \cdot - 12} (3 x^{2} - 12) + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 12$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{x \ln (x)}{x (x^{2} - 12)} (3 x^{2} - 12) + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.2

Cancel the common factor of $x$ .

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Step 13.1.1.2.1

Cancel the common factor.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{x \ln (x)}{x (x^{2} - 12)} (3 x^{2} - 12) + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.2.2

Rewrite the expression.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x)}{x^{2} - 12} (3 x^{2} - 12) + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.3

Multiply $\frac{\ln (x)}{x^{2} - 12}$ by $3 x^{2} - 12$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x) (3 x^{2} - 12)}{x^{2} - 12} + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.4

Simplify the numerator.

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Step 13.1.1.4.1

Factor $3$ out of $3 x^{2} - 12$ .

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Step 13.1.1.4.1.1

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

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x) (3 (x^{2}) - 12)}{x^{2} - 12} + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.4.1.2

Factor $3$ out of $- 12$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x) (3 x^{2} + 3 \cdot - 4)}{x^{2} - 12} + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.4.1.3

Factor $3$ out of $3 x^{2} + 3 \cdot - 4$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x) (3 (x^{2} - 4))}{x^{2} - 12} + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.4.2

Rewrite $4$ as $2^{2}$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x) (3 (x^{2} - 2^{2}))}{x^{2} - 12} + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.4.3

Factor.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x) \cdot 3 (x + 2) (x - 2)}{x^{2} - 12} + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.4.4

Combine exponents.

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Step 13.1.1.4.4.1

Reorder $\ln (x)$ and $3$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{3 \cdot \ln (x) (x + 2) (x - 2)}{x^{2} - 12} + \ln (x^{3} - 12 x))}{x}$

Step 13.1.1.4.4.2

Simplify $3 \cdot \ln (x)$ by moving $3$ inside the logarithm.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x^{3}) (x + 2) (x - 2)}{x^{2} - 12} + \ln (x^{3} - 12 x))}{x}$

Step 13.1.2

To write $\ln (x^{3} - 12 x)$ as a fraction with a common denominator, multiply by $\frac{x^{2} - 12}{x^{2} - 12}$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\frac{\ln (x^{3}) (x + 2) (x - 2)}{x^{2} - 12} + \frac{\ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12})}{x}$

Step 13.1.3

Combine the numerators over the common denominator.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) (x + 2) (x - 2) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4

Simplify the numerator.

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Step 13.1.4.1

Apply the distributive property.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{(\ln (x^{3}) x + \ln (x^{3}) \cdot 2) (x - 2) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.2

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

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Step 13.1.4.2.1

Reorder $\ln (x^{3})$ and $2$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{(\ln (x^{3}) x + 2 \cdot \ln (x^{3})) (x - 2) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.2.2

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

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{(\ln (x^{3}) x + \ln ({(x^{3})}^{2})) (x - 2) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.3

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

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Step 13.1.4.3.1

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

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{(\ln (x^{3}) x + \ln (x^{3 \cdot 2})) (x - 2) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.3.2

Multiply $3$ by $2$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{(\ln (x^{3}) x + \ln (x^{6})) (x - 2) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.4

Expand $(\ln (x^{3}) x + \ln (x^{6})) (x - 2)$ using the FOIL Method.

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Step 13.1.4.4.1

Apply the distributive property.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x (x - 2) + \ln (x^{6}) (x - 2) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.4.2

Apply the distributive property.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x \cdot x + \ln (x^{3}) x \cdot - 2 + \ln (x^{6}) (x - 2) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.4.3

Apply the distributive property.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x \cdot x + \ln (x^{3}) x \cdot - 2 + \ln (x^{6}) x + \ln (x^{6}) \cdot - 2 + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5

Simplify and combine like terms.

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Step 13.1.4.5.1

Simplify each term.

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Step 13.1.4.5.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 13.1.4.5.1.1.1

Move $x$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) (x \cdot x) + \ln (x^{3}) x \cdot - 2 + \ln (x^{6}) x + \ln (x^{6}) \cdot - 2 + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.1.1.2

Multiply $x$ by $x$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} + \ln (x^{3}) x \cdot - 2 + \ln (x^{6}) x + \ln (x^{6}) \cdot - 2 + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.1.2

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

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Step 13.1.4.5.1.2.1

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

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} + x \cdot (- 2 \cdot \ln (x^{3})) + \ln (x^{6}) x + \ln (x^{6}) \cdot - 2 + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.1.2.2

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

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} + x \cdot (- \ln ({(x^{3})}^{2})) + \ln (x^{6}) x + \ln (x^{6}) \cdot - 2 + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.1.3

Rewrite using the commutative property of multiplication.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - x \ln ({(x^{3})}^{2}) + \ln (x^{6}) x + \ln (x^{6}) \cdot - 2 + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.1.4

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

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Step 13.1.4.5.1.4.1

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

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - x \ln (x^{3 \cdot 2}) + \ln (x^{6}) x + \ln (x^{6}) \cdot - 2 + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.1.4.2

Multiply $3$ by $2$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - x \ln (x^{6}) + \ln (x^{6}) x + \ln (x^{6}) \cdot - 2 + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.1.5

Multiply $\ln (x^{6}) \cdot - 2$ .

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Step 13.1.4.5.1.5.1

Reorder $\ln (x^{6})$ and $- 2$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - x \ln (x^{6}) + \ln (x^{6}) x - 2 \cdot \ln (x^{6}) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.1.5.2

Simplify $- 2 \cdot \ln (x^{6})$ by moving $2$ inside the logarithm.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - x \ln (x^{6}) + \ln (x^{6}) x - \ln ({(x^{6})}^{2}) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.1.6

Multiply the exponents in ${(x^{6})}^{2}$ .

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Step 13.1.4.5.1.6.1

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

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - x \ln (x^{6}) + \ln (x^{6}) x - \ln (x^{6 \cdot 2}) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.1.6.2

Multiply $6$ by $2$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - x \ln (x^{6}) + \ln (x^{6}) x - \ln (x^{12}) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.2

Add $- x \ln (x^{6})$ and $\ln (x^{6}) x$ .

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Step 13.1.4.5.2.1

Reorder $\ln (x^{6})$ and $x$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - x \ln (x^{6}) + x \ln (x^{6}) - \ln (x^{12}) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.2.2

Add $- x \ln (x^{6})$ and $x \ln (x^{6})$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} + 0 - \ln (x^{12}) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.5.3

Add $\ln (x^{3}) x^{2}$ and $0$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - \ln (x^{12}) + \ln (x^{3} - 12 x) (x^{2} - 12)}{x^{2} - 12}}{x}$

Step 13.1.4.6

Apply the distributive property.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - \ln (x^{12}) + \ln (x^{3} - 12 x) x^{2} + \ln (x^{3} - 12 x) \cdot - 12}{x^{2} - 12}}{x}$

Step 13.1.4.7

Multiply $\ln (x^{3} - 12 x) \cdot - 12$ .

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Step 13.1.4.7.1

Reorder $\ln (x^{3} - 12 x)$ and $- 12$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - \ln (x^{12}) + \ln (x^{3} - 12 x) x^{2} - 12 \cdot \ln (x^{3} - 12 x)}{x^{2} - 12}}{x}$

Step 13.1.4.7.2

Simplify $- 12 \cdot \ln (x^{3} - 12 x)$ by moving $12$ inside the logarithm.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} \frac{\ln (x^{3}) x^{2} - \ln (x^{12}) + \ln (x^{3} - 12 x) x^{2} - \ln ({(x^{3} - 12 x)}^{12})}{x^{2} - 12}}{x}$

Step 13.1.5

Combine $e^{\ln (x) \ln (x^{3} - 12 x)}$ and $\frac{\ln (x^{3}) x^{2} - \ln (x^{12}) + \ln (x^{3} - 12 x) x^{2} - \ln ({(x^{3} - 12 x)}^{12})}{x^{2} - 12}$ .

$\frac{\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\ln (x^{3}) x^{2} - \ln (x^{12}) + \ln (x^{3} - 12 x) x^{2} - \ln ({(x^{3} - 12 x)}^{12}))}{x^{2} - 12}}{x}$

Step 13.1.6

Reorder factors in $\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (\ln (x^{3}) x^{2} - \ln (x^{12}) + \ln (x^{3} - 12 x) x^{2} - \ln ({(x^{3} - 12 x)}^{12}))}{x^{2} - 12}$ .

$\frac{\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (x^{2} \ln (x^{3}) - \ln (x^{12}) + x^{2} \ln (x^{3} - 12 x) - \ln ({(x^{3} - 12 x)}^{12}))}{x^{2} - 12}}{x}$

Step 13.2

Combine terms.

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Step 13.2.1

Rewrite $\frac{\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (x^{2} \ln (x^{3}) - \ln (x^{12}) + x^{2} \ln (x^{3} - 12 x) - \ln ({(x^{3} - 12 x)}^{12}))}{x^{2} - 12}}{x}$ as a product.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (x^{2} \ln (x^{3}) - \ln (x^{12}) + x^{2} \ln (x^{3} - 12 x) - \ln ({(x^{3} - 12 x)}^{12}))}{x^{2} - 12} \cdot \frac{1}{x}$

Step 13.2.2

Multiply $\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (x^{2} \ln (x^{3}) - \ln (x^{12}) + x^{2} \ln (x^{3} - 12 x) - \ln ({(x^{3} - 12 x)}^{12}))}{x^{2} - 12}$ by $\frac{1}{x}$ .

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (x^{2} \ln (x^{3}) - \ln (x^{12}) + x^{2} \ln (x^{3} - 12 x) - \ln ({(x^{3} - 12 x)}^{12}))}{(x^{2} - 12) x}$

Step 13.3

Reorder terms.

$\frac{e^{\ln (x) \ln (x^{3} - 12 x)} (x^{2} \ln (x^{3}) - \ln (x^{12}) + x^{2} \ln (x^{3} - 12 x) - \ln ({(x^{3} - 12 x)}^{12}))}{x (x^{2} - 12)}$