Calculus Examples

$\ln (\cos (2 x))$

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 $g (x) = \cos (2 x)$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

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 $2 \frac{d}{d x} [x]$ .

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

Step 4.2

Multiply $2$ by $- 1$ .

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

Step 4.3

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

$\sec (2 x) (- 2 \sin (2 x) \cdot 1)$

Step 4.4

Multiply $- 2$ by $1$ .

$\sec (2 x) (- 2 \sin (2 x))$

Step 5

Simplify.

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Step 5.1

Reorder the factors of $\sec (2 x) (- 2 \sin (2 x))$ .

$- 2 \sec (2 x) \sin (2 x)$

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Move the negative in front of the fraction.

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Separate fractions.

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

Step 5.8

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

$- (\frac{2}{1} \tan (2 x))$

Step 5.9

Divide $2$ by $1$ .

$- (2 \tan (2 x))$

Step 5.10

Multiply $2$ by $- 1$ .

$- 2 \tan (2 x)$