Calculus Examples

$| 2 x |$

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 $g (x) = 2 x$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 2

Differentiate.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.2

Combine fractions.

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $2$ .

$\frac{4 x}{| 2 x |} \frac{d}{d x} [x]$

Step 2.3

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

$\frac{4 x}{| 2 x |} \cdot 1$

Step 2.4

Multiply $\frac{4 x}{| 2 x |}$ by $1$ .

$\frac{4 x}{| 2 x |}$

Step 3

Simplify.

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Step 3.1

Remove non-negative terms from the absolute value.

$\frac{4 x}{2 | x |}$

Step 3.2

Cancel the common factor of $4$ and $2$ .

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Step 3.2.1

Factor $2$ out of $4 x$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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