Calculus Examples

$\frac{1 - \cos (x)}{1 + \cos (x)}$

Step 1

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

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

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Multiply $\sin (x)$ by $1$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 5

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

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

Step 6

Multiply.

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Step 6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2

Multiply $1 - \cos (x)$ by $1$ .

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Simplify the numerator.

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Step 7.3.1

Combine the opposite terms in $1 \sin (x) + \cos (x) \sin (x) + 1 \sin (x) - \cos (x) \sin (x)$ .

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Step 7.3.1.1

Subtract $\cos (x) \sin (x)$ from $\cos (x) \sin (x)$ .

$\frac{1 \sin (x) + 0 + 1 \sin (x)}{{(1 + \cos (x))}^{2}}$

Step 7.3.1.2

Add $1 \sin (x)$ and $0$ .

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

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

Multiply $\sin (x)$ by $1$ .

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

Step 7.3.2.2

Multiply $\sin (x)$ by $1$ .

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

Step 7.3.3

Add $\sin (x)$ and $\sin (x)$ .

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