Calculus Examples

$x^{2} | x |$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = | x |$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

Add $2$ and $1$ .

$\frac{x^{3}}{| x |} + | x | \frac{d}{d x} [x^{2}]$

Step 5

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

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

Step 6

Reorder terms.

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