Calculus Examples

$f (x) = \frac{2 \sin (x)}{3 + \cos (x)}$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 \sin (x)}{3 + \cos (x)}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{\sin (x)}{3 + \cos (x)}]$ .

$2 \frac{d}{d x} [\frac{\sin (x)}{3 + \cos (x)}]$

Step 2

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 $g (x) = 3 + \cos (x)$ .

$2 \frac{(3 + \cos (x)) \frac{d}{d x} [\sin (x)] - \sin (x) \frac{d}{d x} [3 + \cos (x)]}{{(3 + \cos (x))}^{2}}$

Step 3

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

$2 \frac{(3 + \cos (x)) \cos (x) - \sin (x) \frac{d}{d x} [3 + \cos (x)]}{{(3 + \cos (x))}^{2}}$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $3 + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [\cos (x)]$ .

$2 \frac{(3 + \cos (x)) \cos (x) - \sin (x) (\frac{d}{d x} [3] + \frac{d}{d x} [\cos (x)])}{{(3 + \cos (x))}^{2}}$

Step 4.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 \frac{(3 + \cos (x)) \cos (x) - \sin (x) (0 + \frac{d}{d x} [\cos (x)])}{{(3 + \cos (x))}^{2}}$

Step 4.3

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

$2 \frac{(3 + \cos (x)) \cos (x) - \sin (x) \frac{d}{d x} [\cos (x)]}{{(3 + \cos (x))}^{2}}$

Step 5

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

$2 \frac{(3 + \cos (x)) \cos (x) - \sin (x) (- \sin (x))}{{(3 + \cos (x))}^{2}}$

Step 6

Multiply.

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Step 6.1

Multiply $- 1$ by $- 1$ .

$2 \frac{(3 + \cos (x)) \cos (x) + 1 \sin (x) \sin (x)}{{(3 + \cos (x))}^{2}}$

Step 6.2

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

$2 \frac{(3 + \cos (x)) \cos (x) + \sin (x) \sin (x)}{{(3 + \cos (x))}^{2}}$

Step 7

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

$2 \frac{(3 + \cos (x)) \cos (x) + \sin^{1} (x) \sin (x)}{{(3 + \cos (x))}^{2}}$

Step 8

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

$2 \frac{(3 + \cos (x)) \cos (x) + \sin^{1} (x) \sin^{1} (x)}{{(3 + \cos (x))}^{2}}$

Step 9

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

$2 \frac{(3 + \cos (x)) \cos (x) + {\sin (x)}^{1 + 1}}{{(3 + \cos (x))}^{2}}$

Step 10

Add $1$ and $1$ .

$2 \frac{(3 + \cos (x)) \cos (x) + \sin^{2} (x)}{{(3 + \cos (x))}^{2}}$

Step 11

Combine $2$ and $\frac{(3 + \cos (x)) \cos (x) + \sin^{2} (x)}{{(3 + \cos (x))}^{2}}$ .

$\frac{2 ((3 + \cos (x)) \cos (x) + \sin^{2} (x))}{{(3 + \cos (x))}^{2}}$

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

$\frac{2 (3 \cos (x) + \cos (x) \cos (x) + \sin^{2} (x))}{{(3 + \cos (x))}^{2}}$

Step 12.2

Apply the distributive property.

$\frac{2 (3 \cos (x)) + 2 (\cos (x) \cos (x)) + 2 \sin^{2} (x)}{{(3 + \cos (x))}^{2}}$

Step 12.3

Simplify the numerator.

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Step 12.3.1

Simplify each term.

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Step 12.3.1.1

Multiply $3$ by $2$ .

$\frac{6 \cos (x) + 2 (\cos (x) \cos (x)) + 2 \sin^{2} (x)}{{(3 + \cos (x))}^{2}}$

Step 12.3.1.2

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

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Step 12.3.1.2.1

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

$\frac{6 \cos (x) + 2 (\cos^{1} (x) \cos (x)) + 2 \sin^{2} (x)}{{(3 + \cos (x))}^{2}}$

Step 12.3.1.2.2

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

$\frac{6 \cos (x) + 2 (\cos^{1} (x) \cos^{1} (x)) + 2 \sin^{2} (x)}{{(3 + \cos (x))}^{2}}$

Step 12.3.1.2.3

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

$\frac{6 \cos (x) + 2 {\cos (x)}^{1 + 1} + 2 \sin^{2} (x)}{{(3 + \cos (x))}^{2}}$

Step 12.3.1.2.4

Add $1$ and $1$ .

$\frac{6 \cos (x) + 2 \cos^{2} (x) + 2 \sin^{2} (x)}{{(3 + \cos (x))}^{2}}$

Step 12.3.2

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

$\frac{6 \cos (x) + 2 (\cos^{2} (x)) + 2 \sin^{2} (x)}{{(3 + \cos (x))}^{2}}$

Step 12.3.3

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

$\frac{6 \cos (x) + 2 (\cos^{2} (x)) + 2 (\sin^{2} (x))}{{(3 + \cos (x))}^{2}}$

Step 12.3.4

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

$\frac{6 \cos (x) + 2 (\cos^{2} (x) + \sin^{2} (x))}{{(3 + \cos (x))}^{2}}$

Step 12.3.5

Rearrange terms.

$\frac{6 \cos (x) + 2 (\sin^{2} (x) + \cos^{2} (x))}{{(3 + \cos (x))}^{2}}$

Step 12.3.6

Apply pythagorean identity.

$\frac{6 \cos (x) + 2 \cdot 1}{{(3 + \cos (x))}^{2}}$

Step 12.3.7

Multiply $2$ by $1$ .

$\frac{6 \cos (x) + 2}{{(3 + \cos (x))}^{2}}$

Step 12.4

Reorder terms.

$\frac{2 + 6 \cos (x)}{{(3 + \cos (x))}^{2}}$

Step 12.5

Factor $2$ out of $2 + 6 \cos (x)$ .

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Step 12.5.1

Factor $2$ out of $2$ .

$\frac{2 (1) + 6 \cos (x)}{{(3 + \cos (x))}^{2}}$

Step 12.5.2

Factor $2$ out of $6 \cos (x)$ .

$\frac{2 (1) + 2 (3 \cos (x))}{{(3 + \cos (x))}^{2}}$

Step 12.5.3

Factor $2$ out of $2 (1) + 2 (3 \cos (x))$ .

$\frac{2 (1 + 3 \cos (x))}{{(3 + \cos (x))}^{2}}$