Calculus Examples

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

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 $f (x) = \sin (x)$ and $g (x) = 2 - \cos (x)$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply.

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Step 3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Simplify the numerator.

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Step 9.2.1

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

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Step 9.2.1.1

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

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

Step 9.2.1.2

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

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

Step 9.2.1.3

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

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

Step 9.2.1.4

Add $1$ and $1$ .

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

Step 9.2.2

Factor $- 1$ out of $- \cos^{2} (x)$ .

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

Step 9.2.3

Factor $- 1$ out of $- \sin^{2} (x)$ .

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

Step 9.2.4

Factor $- 1$ out of $- (\cos^{2} (x)) - (\sin^{2} (x))$ .

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

Step 9.2.5

Rearrange terms.

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

Step 9.2.6

Apply pythagorean identity.

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

Step 9.2.7

Multiply $- 1$ by $1$ .

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

Step 9.3

Reorder terms.

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

Step 9.4

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 9.5

Factor $- 1$ out of $2 \cos (x)$ .

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

Step 9.6

Factor $- 1$ out of $- 1 (1) - (- 2 \cos (x))$ .

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

Step 9.7

Move the negative in front of the fraction.

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