Calculus Examples

$y = 5 \ln^{4} (x)$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 \ln^{4} (x)$ with respect to $x$ is $5 \frac{d}{d x} [\ln^{4} (x)]$ .

$5 \frac{d}{d x} [\ln^{4} (x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \ln (x)$ .

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Step 2.1

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

$5 (\frac{d}{d u} [u^{4}] \frac{d}{d x} [\ln (x)])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$5 (4 u^{3} \frac{d}{d x} [\ln (x)])$

Step 2.3

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

$5 (4 \ln^{3} (x) \frac{d}{d x} [\ln (x)])$

Step 3

Multiply $4$ by $5$ .

$20 (\ln^{3} (x) \frac{d}{d x} [\ln (x)])$

Step 4

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

$20 \ln^{3} (x) \frac{1}{x}$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{x}$ and $20$ .

$\frac{20}{x} \ln^{3} (x)$

Step 5.2

Combine $\frac{20}{x}$ and $\ln^{3} (x)$ .

$\frac{20 \ln^{3} (x)}{x}$