Calculus Examples

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

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

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

Step 2

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 $f (x) = x$ and $g (x) = e^{x}$ .

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

Multiply $e^{x}$ by $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Differentiate.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

Combine fractions.

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Step 6.6.1

Add $2 x$ and $0$ .

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

Step 6.6.2

Multiply $2$ by $- 1$ .

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

Step 6.6.3

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

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

Step 6.6.4

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Add $1$ and $1$ .

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

Step 11

Move the negative in front of the fraction.

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

Step 12

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

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

Step 13

Combine the numerators over the common denominator.

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Simplify each term.

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Step 16.1.1.1

Apply the distributive property.

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

Step 16.1.1.2

Expand $(\ln (x^{2} - 1) (x e^{x}) + \ln (x^{2} - 1) e^{x}) (x^{2} - 1)$ using the FOIL Method.

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Step 16.1.1.2.1

Apply the distributive property.

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

Step 16.1.1.2.2

Apply the distributive property.

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

Step 16.1.1.2.3

Apply the distributive property.

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

Step 16.1.1.3

Simplify each term.

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Step 16.1.1.3.1

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

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Step 16.1.1.3.1.1

Move $x^{2}$ .

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

Step 16.1.1.3.1.2

Multiply $x^{2}$ by $x$ .

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Step 16.1.1.3.1.2.1

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

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

Step 16.1.1.3.1.2.2

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

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

Step 16.1.1.3.1.3

Add $2$ and $1$ .

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

Step 16.1.1.3.2

Move $- 1$ to the left of $\ln (x^{2} - 1) (x e^{x})$ .

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

Step 16.1.1.3.3

Rewrite $- 1 (\ln (x^{2} - 1) (x e^{x}))$ as $- (\ln (x^{2} - 1) (x e^{x}))$ .

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

Step 16.1.1.3.4

Move $- 1$ to the left of $\ln (x^{2} - 1) e^{x}$ .

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

Step 16.1.1.3.5

Rewrite $- 1 (\ln (x^{2} - 1) e^{x})$ as $- (\ln (x^{2} - 1) e^{x})$ .

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

Step 16.1.2

Reorder factors in $\ln (x^{2} - 1) (x^{3} e^{x}) - \ln (x^{2} - 1) (x e^{x}) + \ln (x^{2} - 1) e^{x} x^{2} - \ln (x^{2} - 1) e^{x} - 2 e^{x} x^{2}$ .

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

Step 16.2

Reorder terms.

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