Calculus Examples

$\ln (| x |)$

Step 1

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

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Step 1.1

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

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

Step 1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [| x |]$

Step 1.3

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

$\frac{1}{| x |} \frac{d}{d x} [| x |]$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

Simplify.

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Step 9.1

Remove non-negative terms from the absolute value.

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

Step 9.2

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

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Step 9.2.1

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

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

Step 9.2.2

Factor $x$ out of $x^{1}$ .

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

Step 9.2.3

Cancel the common factors.

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Step 9.2.3.1

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

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

Step 9.2.3.2

Cancel the common factor.

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

Step 9.2.3.3

Rewrite the expression.

$\frac{1}{x}$