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

Differentiate using the Power Rule.

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Step 2.1

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

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

Step 2.2

Multiply $| x |$ by $1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

Simplify.

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Step 9.1

Reorder terms.

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

Step 9.2

Simplify the numerator.

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Step 9.2.1

To write $| x |$ as a fraction with a common denominator, multiply by $\frac{| x |}{| x |}$ .

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

Step 9.2.2

Combine the numerators over the common denominator.

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

Step 9.2.3

Simplify the numerator.

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Step 9.2.3.1

Multiply $| x | | x |$ .

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Step 9.2.3.1.1

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

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

Step 9.2.3.1.2

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

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

Step 9.2.3.1.3

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

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

Step 9.2.3.1.4

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

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

Step 9.2.3.1.5

Add $1$ and $1$ .

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

Step 9.2.3.2

Remove non-negative terms from the absolute value.

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

Step 9.2.3.3

Add $- x^{2}$ and $x^{2}$ .

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

Step 9.2.4

Divide $0$ by $| x |$ .

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

Step 9.3

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

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

Step 9.4

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

$0$