Calculus Examples

$\frac{1}{| x |}$

Step 1

Rewrite $\frac{1}{| x |}$ as ${| x |}^{- 1}$ .

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

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 $g (x) = | x |$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $| x |$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $| x |$ .

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

Step 3

The derivative of $| x |$ with respect to $x$ is $\frac{x}{| x |}$ .

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

Move ${| x |}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5

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

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

Add $1$ and $2$ .

$- \frac{x}{{| x |}^{3}}$