Calculus Examples

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

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

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $2 x$ and $0$ .

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

Step 3.4.2

Multiply $2$ by $- 1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

Rewrite $- 1 e^{x}$ as $- e^{x}$ .

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

Step 4.2.2

Reorder factors in $x^{2} e^{x} - e^{x} - 2 e^{x} x$ .

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

Step 4.3

Reorder terms.

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