Calculus Examples

$\frac{4 - \cos (x)}{4 + \sin (x)}$

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 $f (x) = 4 - \cos (x)$ and $g (x) = 4 + \sin (x)$ .

$\frac{(4 + \sin (x)) \frac{d}{d x} [4 - \cos (x)] - (4 - \cos (x)) \frac{d}{d x} [4 + \sin (x)]}{{(4 + \sin (x))}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- \cos (x)]$ .

$\frac{(4 + \sin (x)) (\frac{d}{d x} [4] + \frac{d}{d x} [- \cos (x)]) - (4 - \cos (x)) \frac{d}{d x} [4 + \sin (x)]}{{(4 + \sin (x))}^{2}}$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{(4 + \sin (x)) (0 + \frac{d}{d x} [- \cos (x)]) - (4 - \cos (x)) \frac{d}{d x} [4 + \sin (x)]}{{(4 + \sin (x))}^{2}}$

Step 2.3

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

$\frac{(4 + \sin (x)) \frac{d}{d x} [- \cos (x)] - (4 - \cos (x)) \frac{d}{d x} [4 + \sin (x)]}{{(4 + \sin (x))}^{2}}$

Step 2.4

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

$\frac{(4 + \sin (x)) (- \frac{d}{d x} [\cos (x)]) - (4 - \cos (x)) \frac{d}{d x} [4 + \sin (x)]}{{(4 + \sin (x))}^{2}}$

Step 3

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

$\frac{(4 + \sin (x)) (- - \sin (x)) - (4 - \cos (x)) \frac{d}{d x} [4 + \sin (x)]}{{(4 + \sin (x))}^{2}}$

Step 4

Differentiate.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$\frac{(4 + \sin (x)) (1 \sin (x)) - (4 - \cos (x)) \frac{d}{d x} [4 + \sin (x)]}{{(4 + \sin (x))}^{2}}$

Step 4.2

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

$\frac{(4 + \sin (x)) \sin (x) - (4 - \cos (x)) \frac{d}{d x} [4 + \sin (x)]}{{(4 + \sin (x))}^{2}}$

Step 4.3

By the Sum Rule, the derivative of $4 + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [\sin (x)]$ .

$\frac{(4 + \sin (x)) \sin (x) - (4 - \cos (x)) (\frac{d}{d x} [4] + \frac{d}{d x} [\sin (x)])}{{(4 + \sin (x))}^{2}}$

Step 4.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{(4 + \sin (x)) \sin (x) - (4 - \cos (x)) (0 + \frac{d}{d x} [\sin (x)])}{{(4 + \sin (x))}^{2}}$

Step 4.5

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

$\frac{(4 + \sin (x)) \sin (x) - (4 - \cos (x)) \frac{d}{d x} [\sin (x)]}{{(4 + \sin (x))}^{2}}$

Step 5

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

$\frac{(4 + \sin (x)) \sin (x) - (4 - \cos (x)) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{4 \sin (x) + \sin (x) \sin (x) - (4 - \cos (x)) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.2

Apply the distributive property.

$\frac{4 \sin (x) + \sin (x) \sin (x) + (- 1 \cdot 4 - - \cos (x)) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.3

Apply the distributive property.

$\frac{4 \sin (x) + \sin (x) \sin (x) - 1 \cdot 4 \cos (x) - - \cos (x) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

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

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Step 6.4.1.1.1

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

$\frac{4 \sin (x) + \sin^{1} (x) \sin (x) - 1 \cdot 4 \cos (x) - - \cos (x) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.1.1.2

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

$\frac{4 \sin (x) + \sin^{1} (x) \sin^{1} (x) - 1 \cdot 4 \cos (x) - - \cos (x) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.1.1.3

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

$\frac{4 \sin (x) + {\sin (x)}^{1 + 1} - 1 \cdot 4 \cos (x) - - \cos (x) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.1.1.4

Add $1$ and $1$ .

$\frac{4 \sin (x) + \sin^{2} (x) - 1 \cdot 4 \cos (x) - - \cos (x) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.1.2

Multiply $- 1$ by $4$ .

$\frac{4 \sin (x) + \sin^{2} (x) - 4 \cos (x) - - \cos (x) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.1.3

Multiply $- - \cos (x)$ .

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Step 6.4.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{4 \sin (x) + \sin^{2} (x) - 4 \cos (x) + 1 \cos (x) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.1.3.2

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

$\frac{4 \sin (x) + \sin^{2} (x) - 4 \cos (x) + \cos (x) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.1.4

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

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Step 6.4.1.4.1

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

$\frac{4 \sin (x) + \sin^{2} (x) - 4 \cos (x) + \cos^{1} (x) \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.1.4.2

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

$\frac{4 \sin (x) + \sin^{2} (x) - 4 \cos (x) + \cos^{1} (x) \cos^{1} (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.1.4.3

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

$\frac{4 \sin (x) + \sin^{2} (x) - 4 \cos (x) + {\cos (x)}^{1 + 1}}{{(4 + \sin (x))}^{2}}$

Step 6.4.1.4.4

Add $1$ and $1$ .

$\frac{4 \sin (x) + \sin^{2} (x) - 4 \cos (x) + \cos^{2} (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.2

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

$\frac{4 \sin (x) + \sin^{2} (x) + \cos^{2} (x) - 4 \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.4.3

Apply pythagorean identity.

$\frac{4 \sin (x) + 1 - 4 \cos (x)}{{(4 + \sin (x))}^{2}}$

Step 6.5

Reorder terms.

$\frac{1 + 4 \sin (x) - 4 \cos (x)}{{(4 + \sin (x))}^{2}}$