Calculus Examples

$f (x) = 2 \ln (\sin (x))$

Step 1

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

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

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

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Step 2.1

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

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

Step 2.2

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

$2 (\frac{1}{u} \frac{d}{d x} [\sin (x)])$

Step 2.3

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

$2 (\frac{1}{\sin (x)} \frac{d}{d x} [\sin (x)])$

Step 3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$2 (\csc (x) \frac{d}{d x} [\sin (x)])$

Step 4

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

$2 \csc (x) \cos (x)$

Step 5

Simplify.

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Step 5.1

Reorder the factors of $2 \csc (x) \cos (x)$ .

$2 \cos (x) \csc (x)$

Step 5.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$2 \cos (x) \frac{1}{\sin (x)}$

Step 5.3

Multiply $2 \cos (x) \frac{1}{\sin (x)}$ .

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Step 5.3.1

Combine $\frac{1}{\sin (x)}$ and $2$ .

$\frac{2}{\sin (x)} \cos (x)$

Step 5.3.2

Combine $\frac{2}{\sin (x)}$ and $\cos (x)$ .

$\frac{2 \cos (x)}{\sin (x)}$

Step 5.4

Separate fractions.

$\frac{2}{1} \cdot \frac{\cos (x)}{\sin (x)}$

Step 5.5

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{2}{1} \cot (x)$

Step 5.6

Divide $2$ by $1$ .

$2 \cot (x)$