Calculus Examples

$\sqrt{| x |}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{| x |}$ as ${| x |}^{\frac{1}{2}}$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

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

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

Step 9

Multiply $\frac{1}{2 {| x |}^{\frac{1}{2}}}$ by $\frac{x}{| x |}$ .

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

Step 10

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

$\frac{x}{2 ({| x |}^{1} {| x |}^{\frac{1}{2}})}$

Step 11

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

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

Step 12

Write $1$ as a fraction with a common denominator.

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

Step 13

Combine the numerators over the common denominator.

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

Step 14

Add $2$ and $1$ .

$\frac{x}{2 {| x |}^{\frac{3}{2}}}$