Calculus Examples

$f (x) = {(x - 2 \ln (x))}^{4}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = x - 2 \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x - 2 \ln (x)$ .

$\frac{d}{d u} [u^{4}] \frac{d}{d x} [x - 2 \ln (x)]$

Step 1.2

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

$4 u^{3} \frac{d}{d x} [x - 2 \ln (x)]$

Step 1.3

Replace all occurrences of $u$ with $x - 2 \ln (x)$ .

$4 {(x - 2 \ln (x))}^{3} \frac{d}{d x} [x - 2 \ln (x)]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x - 2 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2 \ln (x)]$ .

$4 {(x - 2 \ln (x))}^{3} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2 \ln (x)])$

Step 2.2

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

$4 {(x - 2 \ln (x))}^{3} (1 + \frac{d}{d x} [- 2 \ln (x)])$

Step 2.3

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

$4 {(x - 2 \ln (x))}^{3} (1 - 2 \frac{d}{d x} [\ln (x)])$

Step 3

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

$4 {(x - 2 \ln (x))}^{3} (1 - 2 \frac{1}{x})$

Step 4

Combine fractions.

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Step 4.1

Combine $- 2$ and $\frac{1}{x}$ .

$4 {(x - 2 \ln (x))}^{3} (1 + \frac{- 2}{x})$

Step 4.2

Move the negative in front of the fraction.

$4 {(x - 2 \ln (x))}^{3} (1 - \frac{2}{x})$