Calculus Examples

$\cos ({(1 - 2 x)}^{2})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = {(1 - 2 x)}^{2}$ .

Tap for more steps...

Step 1.1

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

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

Step 1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with ${(1 - 2 x)}^{2}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 1 - 2 x$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

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

Step 2.3

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

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Multiply $2$ by $- 1$ .

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

Step 3.2

By the Sum Rule, the derivative of $1 - 2 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 2 x]$ .

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

Step 3.3

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

$- 2 \sin ({(1 - 2 x)}^{2}) ((1 - 2 x) (0 + \frac{d}{d x} [- 2 x]))$

Step 3.4

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

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

Step 3.5

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

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

Step 3.6

Multiply $- 2$ by $- 2$ .

$4 \sin ({(1 - 2 x)}^{2}) ((1 - 2 x) (\frac{d}{d x} [x]))$

Step 3.7

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

$4 \sin ({(1 - 2 x)}^{2}) ((1 - 2 x) \cdot 1)$

Step 3.8

Multiply $1 - 2 x$ by $1$ .

$4 \sin ({(1 - 2 x)}^{2}) (1 - 2 x)$