Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $1 - \cos (x)$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $1 - \cos (x)$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Add $0$ and $\frac{d}{d x} [- \cos (x)]$ .

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

Step 2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

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

Step 2.5

Multiply $- 1$ by $2$ .

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

Step 3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 4

Multiply $- 1$ by $- 2$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Combine terms.

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Step 5.3.1

Multiply $2$ by $1$ .

$2 \sin (x) + 2 (- \cos (x)) \sin (x)$

Step 5.3.2

Multiply $- 1$ by $2$ .

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

Step 5.4

Reorder terms.

$- 2 \cos (x) \sin (x) + 2 \sin (x)$