Calculus Examples

$\ln (\cos (w - 1))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = \ln (w)$ and $g (w) = \cos (w - 1)$ .

Tap for more steps...

Step 1.1

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

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d w} [\cos (w - 1)]$

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d w} [\cos (w - 1)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\cos (w - 1)$ .

$\frac{1}{\cos (w - 1)} \frac{d}{d w} [\cos (w - 1)]$

Step 2

Convert from $\frac{1}{\cos (w - 1)}$ to $\sec (w - 1)$ .

$\sec (w - 1) \frac{d}{d w} [\cos (w - 1)]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = \cos (w)$ and $g (w) = w - 1$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u_{2}$ as $w - 1$ .

$\sec (w - 1) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d w} [w - 1])$

Step 3.2

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

$\sec (w - 1) (- \sin (u_{2}) \frac{d}{d w} [w - 1])$

Step 3.3

Replace all occurrences of $u_{2}$ with $w - 1$ .

$\sec (w - 1) (- \sin (w - 1) \frac{d}{d w} [w - 1])$

Step 4

Differentiate.

Tap for more steps...

Step 4.1

By the Sum Rule, the derivative of $w - 1$ with respect to $w$ is $\frac{d}{d w} [w] + \frac{d}{d w} [- 1]$ .

$\sec (w - 1) (- \sin (w - 1) (\frac{d}{d w} [w] + \frac{d}{d w} [- 1]))$

Step 4.2

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

$\sec (w - 1) (- \sin (w - 1) (1 + \frac{d}{d w} [- 1]))$

Step 4.3

Since $- 1$ is constant with respect to $w$ , the derivative of $- 1$ with respect to $w$ is $0$ .

$\sec (w - 1) (- \sin (w - 1) (1 + 0))$

Step 4.4

Simplify the expression.

Tap for more steps...

Step 4.4.1

Add $1$ and $0$ .

$\sec (w - 1) (- \sin (w - 1) \cdot 1)$

Step 4.4.2

Multiply $- 1$ by $1$ .

$\sec (w - 1) (- \sin (w - 1))$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Reorder the factors of $\sec (w - 1) (- \sin (w - 1))$ .

$- \sec (w - 1) \sin (w - 1)$

Step 5.2

Rewrite $\sec (w - 1)$ in terms of sines and cosines.

$- \frac{1}{\cos (w - 1)} \sin (w - 1)$

Step 5.3

Combine $\sin (w - 1)$ and $\frac{1}{\cos (w - 1)}$ .

$- \frac{\sin (w - 1)}{\cos (w - 1)}$

Step 5.4

Convert from $\frac{\sin (w - 1)}{\cos (w - 1)}$ to $\tan (w - 1)$ .

$- \tan (w - 1)$