Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $2$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply $- 2$ by $- 2$ .

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

Step 3.7

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

$4 \cos (1 - 2 x) \sin (1 - 2 x) \cdot 1$

Step 3.8

Multiply $4$ by $1$ .

$4 \cos (1 - 2 x) \sin (1 - 2 x)$