Calculus Examples

$f (x) = 4 \cos (2 \ln (x))$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \cos (2 \ln (x))$ with respect to $x$ is $4 \frac{d}{d x} [\cos (2 \ln (x))]$ .

$4 \frac{d}{d x} [\cos (2 \ln (x))]$

Step 2

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

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Step 2.1

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

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

Step 2.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

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

Step 2.3

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

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

Step 3

Differentiate using the Constant Multiple Rule.

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Step 3.1

Multiply $- 1$ by $4$ .

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

Step 3.2

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

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

Step 3.3

Multiply $2$ by $- 4$ .

$- 8 \sin (2 \ln (x)) \frac{d}{d x} [\ln (x)]$

Step 4

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

$- 8 \sin (2 \ln (x)) \frac{1}{x}$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{x}$ and $- 8$ .

$\frac{- 8}{x} \sin (2 \ln (x))$

Step 5.2

Combine $\frac{- 8}{x}$ and $\sin (2 \ln (x))$ .

$\frac{- 8 \sin (2 \ln (x))}{x}$

Step 5.3

Move the negative in front of the fraction.

$- \frac{8 \sin (2 \ln (x))}{x}$

Step 6

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

$- \frac{8 \sin (\ln (x^{2}))}{x}$