Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (2 x)$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (2 x)$ .

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

Step 2

Convert from $\frac{1}{\sin (2 x)}$ to $\csc (2 x)$ .

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

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

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

Step 3.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

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

Step 3.3

Replace all occurrences of $u_{2}$ with $2 x$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

$\csc (2 x) \cos (2 x) (2 \cdot 1)$

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $2$ by $1$ .

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

Step 4.3.2

Move $2$ to the left of $\csc (2 x) \cos (2 x)$ .

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Rewrite $\csc (2 x)$ in terms of sines and cosines.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.4

Separate fractions.

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

Step 5.5

Convert from $\frac{\cos (2 x)}{\sin (2 x)}$ to $\cot (2 x)$ .

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

Step 5.6

Divide $2$ by $1$ .

$2 \cot (2 x)$