Calculus Examples

ln (sin (x))

$\ln (\sin (x))$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \ln (x)$ and

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

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Step 1.1

To apply the Chain Rule, set

$u$ as

$\sin (x)$ .

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

Step 1.2

The derivative of

$\ln (u)$ with respect to

$u$ is

$\frac{1}{u}$ .

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

Step 1.3

Replace all occurrences of

$u$ with

$\sin (x)$ .

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

Step 2

Convert from

$\frac{1}{\sin (x)}$ to

$\csc (x)$ .

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

Step 3

The derivative of

$\sin (x)$ with respect to

$x$ is

$\cos (x)$ .

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

Step 4

Simplify.

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Step 4.1

Reorder the factors of

$\csc (x) \cos (x)$ .

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

Step 4.2

Rewrite

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

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

Step 4.3

Combine

$\cos (x)$ and

$\frac{1}{\sin (x)}$ .

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

Step 4.4

Convert from

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

$\cot (x)$ .

$\cot (x)$

$\ln (\sin x)$

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