Calculus Examples

$y = 9 \ln (\ln (x))$

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $9 \ln (\ln (x))$ with respect to $x$ is $9 \frac{d}{d x} [\ln (\ln (x))]$ .

$9 \frac{d}{d x} [\ln (\ln (x))]$

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

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Step 2.1

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

$9 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\ln (x)])$

Step 2.2

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

$9 (\frac{1}{u} \frac{d}{d x} [\ln (x)])$

Step 2.3

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

$9 (\frac{1}{\ln (x)} \frac{d}{d x} [\ln (x)])$

Step 3

Combine $\frac{1}{\ln (x)}$ and $9$ .

$\frac{9}{\ln (x)} \frac{d}{d x} [\ln (x)]$

Step 4

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

$\frac{9}{\ln (x)} \cdot \frac{1}{x}$

Step 5

Combine fractions.

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Step 5.1

Multiply $\frac{9}{\ln (x)}$ by $\frac{1}{x}$ .

$\frac{9}{\ln (x) x}$

Step 5.2

Reorder terms.

$\frac{9}{x \ln (x)}$