Calculus Examples

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

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 $f (x) = x^{2}$ and $g (x) = | x |$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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 $1$ .

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

Step 5.1.2

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

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

Step 5.2

Add $1$ and $2$ .

$\frac{| x | (2 x) - \frac{x^{3}}{| x |}}{{| x |}^{2}}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.1.2

Reorder factors in $2 | x | x - \frac{x^{3}}{| x |}$ .

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

Step 6.2

Reorder terms.

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

Step 6.3

Simplify the numerator.

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Step 6.3.1

Factor $x$ out of $- \frac{x^{3}}{| x |} + 2 x | x |$ .

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Step 6.3.1.1

Factor $x$ out of $- \frac{x^{3}}{| x |}$ .

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

Step 6.3.1.2

Factor $x$ out of $2 x | x |$ .

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

Step 6.3.1.3

Factor $x$ out of $x (- \frac{x^{2}}{| x |}) + x (2 | x |)$ .

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

Step 6.3.2

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

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

Step 6.3.3

Combine the numerators over the common denominator.

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

Step 6.3.4

Simplify the numerator.

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Step 6.3.4.1

Multiply $2 | x | | x |$ .

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Step 6.3.4.1.1

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

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

Step 6.3.4.1.2

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

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

Step 6.3.4.1.3

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

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

Step 6.3.4.1.4

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

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

Step 6.3.4.1.5

Add $1$ and $1$ .

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

Step 6.3.4.2

Remove non-negative terms from the absolute value.

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

Step 6.3.4.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

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

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Step 6.6.1

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

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Step 6.6.1.1

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

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

Step 6.6.1.2

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

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

Step 6.6.2

Add $1$ and $2$ .

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

Step 6.7

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{3}}{| x |} \cdot \frac{1}{x^{2}}$

Step 6.8

Combine.

$\frac{x^{3} \cdot 1}{| x | x^{2}}$

Step 6.9

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 6.9.1

Factor $x^{2}$ out of $x^{3} \cdot 1$ .

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

Step 6.9.2

Cancel the common factors.

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Step 6.9.2.1

Factor $x^{2}$ out of $| x | x^{2}$ .

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

Step 6.9.2.2

Cancel the common factor.

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

Step 6.9.2.3

Rewrite the expression.

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

Step 6.10

Multiply $x$ by $1$ .

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