Calculus Examples

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

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 $g (x) = \sin (x)$ .

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

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

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

Step 4

Multiply.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

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

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

Step 5

Raise $\sin (x)$ to the power of $1$ .

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

Step 6

Raise $\sin (x)$ to the power of $1$ .

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

Step 7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8

Add $1$ and $1$ .

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Apply the distributive property.

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

Step 10.3

Simplify the numerator.

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Step 10.3.1

Simplify each term.

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Step 10.3.1.1

Multiply $- 1$ by $1$ .

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

Step 10.3.1.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 10.3.1.3

Multiply $- - \cos (x)$ .

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Step 10.3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 10.3.1.3.2

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

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

Step 10.3.1.4

Multiply $\cos (x) \cos (x)$ .

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Step 10.3.1.4.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 10.3.1.4.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 10.3.1.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10.3.1.4.4

Add $1$ and $1$ .

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

Step 10.3.2

Move $\cos^{2} (x)$ .

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

Step 10.3.3

Apply pythagorean identity.

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