Calculus Examples

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

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 $f (x) = | x |$ and $g (x) = x$ .

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

Step 2

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

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

Step 3

Combine $x$ and $\frac{x}{| x |}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

Combine.

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

Step 10

Apply the distributive property.

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

Step 11

Cancel the common factor of $| x |$ .

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Step 11.1

Cancel the common factor.

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

Step 11.2

Rewrite the expression.

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

Step 12

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Add $1$ and $1$ .

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

Step 17

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

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

Step 18

Multiply $- 1$ by $1$ .

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

Step 19

Simplify.

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Step 19.1

Simplify the numerator.

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Step 19.1.1

Remove non-negative terms from the absolute value.

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

Step 19.1.2

Subtract $x^{2}$ from $x^{2}$ .

$\frac{0}{| x | x^{2}}$

Step 19.2

Reorder terms.

$\frac{0}{x^{2} | x |}$

Step 19.3

Divide $0$ by $x^{2} | x |$ .

$0$