Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 1 - x^{2}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $2$ .

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

Step 3.2

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

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

Step 3.3

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

$- 2 {(1 - x^{2})}^{- 2} (0 + \frac{d}{d x} [- x^{2}])$

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply $- 1$ by $- 2$ .

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

Step 3.7

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

$2 {(1 - x^{2})}^{- 2} (2 x)$

Step 3.8

Multiply $2$ by $2$ .

$4 {(1 - x^{2})}^{- 2} x$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 \frac{1}{{(1 - x^{2})}^{2}} x$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

Combine $4$ and $\frac{1}{{(1 - x^{2})}^{2}}$ .

$\frac{4}{{(1 - x^{2})}^{2}} x$

Step 5.1.2

Combine $\frac{4}{{(1 - x^{2})}^{2}}$ and $x$ .

$\frac{4 x}{{(1 - x^{2})}^{2}}$

Step 5.2

Reorder terms.

$\frac{4 x}{{(- x^{2} + 1)}^{2}}$