Calculus Examples

$| x^{2} |$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $x^{2}$ .

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

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 $n = 2$ .

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

Step 2.2

Combine fractions.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Add $1$ and $2$ .

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

Step 6

Simplify.

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Step 6.1

Remove non-negative terms from the absolute value.

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

Step 6.2

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

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Step 6.2.1

Factor $x^{2}$ out of $2 x^{3}$ .

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

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Multiply by $1$ .

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

Step 6.2.2.2

Cancel the common factor.

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

Step 6.2.2.3

Rewrite the expression.

$\frac{2 x}{1}$

Step 6.2.2.4

Divide $2 x$ by $1$ .

$2 x$