Calculus Examples

$3 \ln (11 \ln (x))$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 \ln (11 \ln (x))$ with respect to $x$ is $3 \frac{d}{d x} [\ln (11 \ln (x))]$ .

$3 \frac{d}{d x} [\ln (11 \ln (x))]$

Step 2

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 $g (x) = 11 \ln (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $11 \ln (x)$ .

$3 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [11 \ln (x)])$

Step 2.2

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

$3 (\frac{1}{u} \frac{d}{d x} [11 \ln (x)])$

Step 2.3

Replace all occurrences of $u$ with $11 \ln (x)$ .

$3 (\frac{1}{11 \ln (x)} \frac{d}{d x} [11 \ln (x)])$

Step 3

Differentiate using the Constant Multiple Rule.

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Step 3.1

Combine $\frac{1}{11 \ln (x)}$ and $3$ .

$\frac{3}{11 \ln (x)} \frac{d}{d x} [11 \ln (x)]$

Step 3.2

Since $11$ is constant with respect to $x$ , the derivative of $11 \ln (x)$ with respect to $x$ is $11 \frac{d}{d x} [\ln (x)]$ .

$\frac{3}{11 \ln (x)} (11 \frac{d}{d x} [\ln (x)])$

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $11$ and $\frac{3}{11 \ln (x)}$ .

$\frac{11 \cdot 3}{11 \ln (x)} \frac{d}{d x} [\ln (x)]$

Step 3.3.2

Multiply $11$ by $3$ .

$\frac{33}{11 \ln (x)} \frac{d}{d x} [\ln (x)]$

Step 3.3.3

Cancel the common factor of $33$ and $11$ .

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Step 3.3.3.1

Factor $11$ out of $33$ .

$\frac{11 \cdot 3}{11 \ln (x)} \frac{d}{d x} [\ln (x)]$

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $11$ out of $11 \ln (x)$ .

$\frac{11 \cdot 3}{11 (\ln (x))} \frac{d}{d x} [\ln (x)]$

Step 3.3.3.2.2

Cancel the common factor.

$\frac{11 \cdot 3}{11 \ln (x)} \frac{d}{d x} [\ln (x)]$

Step 3.3.3.2.3

Rewrite the expression.

$\frac{3}{\ln (x)} \frac{d}{d x} [\ln (x)]$

Step 4

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

$\frac{3}{\ln (x)} \cdot \frac{1}{x}$

Step 5

Combine fractions.

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Step 5.1

Multiply $\frac{3}{\ln (x)}$ by $\frac{1}{x}$ .

$\frac{3}{\ln (x) x}$

Step 5.2

Reorder terms.

$\frac{3}{x \ln (x)}$