Calculus Examples

$f (x) = \frac{5 (1 - \sin (x))}{2 \cos (x)}$

Step 1

Since $\frac{5}{2}$ is constant with respect to $x$ , the derivative of $\frac{5 (1 - \sin (x))}{2 \cos (x)}$ with respect to $x$ is $\frac{5}{2} \frac{d}{d x} [\frac{1 - \sin (x)}{\cos (x)}]$ .

$\frac{5}{2} \frac{d}{d x} [\frac{1 - \sin (x)}{\cos (x)}]$

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 $f (x) = 1 - \sin (x)$ and $g (x) = \cos (x)$ .

$\frac{5}{2} \cdot \frac{\cos (x) \frac{d}{d x} [1 - \sin (x)] - (1 - \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 3

Differentiate.

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Step 3.1

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

$\frac{5}{2} \cdot \frac{\cos (x) (\frac{d}{d x} [1] + \frac{d}{d x} [- \sin (x)]) - (1 - \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 3.2

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

$\frac{5}{2} \cdot \frac{\cos (x) (0 + \frac{d}{d x} [- \sin (x)]) - (1 - \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 3.3

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

$\frac{5}{2} \cdot \frac{\cos (x) \frac{d}{d x} [- \sin (x)] - (1 - \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 3.4

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

$\frac{5}{2} \cdot \frac{\cos (x) (- \frac{d}{d x} [\sin (x)]) - (1 - \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 4

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

$\frac{5}{2} \cdot \frac{\cos (x) (- \cos (x)) - (1 - \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 5

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

$\frac{5}{2} \cdot \frac{- (\cos^{1} (x) \cos (x)) - (1 - \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 6

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

$\frac{5}{2} \cdot \frac{- (\cos^{1} (x) \cos^{1} (x)) - (1 - \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 7

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

$\frac{5}{2} \cdot \frac{- {\cos (x)}^{1 + 1} - (1 - \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 8

Add $1$ and $1$ .

$\frac{5}{2} \cdot \frac{- \cos^{2} (x) - (1 - \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 9

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

$\frac{5}{2} \cdot \frac{- \cos^{2} (x) - (1 - \sin (x)) (- \sin (x))}{\cos^{2} (x)}$

Step 10

Combine fractions.

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Step 10.1

Multiply $- 1$ by $- 1$ .

$\frac{5}{2} \cdot \frac{- \cos^{2} (x) + 1 (1 - \sin (x)) \sin (x)}{\cos^{2} (x)}$

Step 10.2

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

$\frac{5}{2} \cdot \frac{- \cos^{2} (x) + (1 - \sin (x)) \sin (x)}{\cos^{2} (x)}$

Step 10.3

Multiply $\frac{5}{2}$ by $\frac{- \cos^{2} (x) + (1 - \sin (x)) \sin (x)}{\cos^{2} (x)}$ .

$\frac{5 (- \cos^{2} (x) + (1 - \sin (x)) \sin (x))}{2 \cos^{2} (x)}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$\frac{5 (- \cos^{2} (x) + 1 \sin (x) - \sin (x) \sin (x))}{2 \cos^{2} (x)}$

Step 11.2

Apply the distributive property.

$\frac{5 (- \cos^{2} (x)) + 5 (1 \sin (x)) + 5 (- \sin (x) \sin (x))}{2 \cos^{2} (x)}$

Step 11.3

Simplify the numerator.

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Step 11.3.1

Simplify each term.

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Step 11.3.1.1

Multiply $- 1$ by $5$ .

$\frac{- 5 \cos^{2} (x) + 5 (1 \sin (x)) + 5 (- \sin (x) \sin (x))}{2 \cos^{2} (x)}$

Step 11.3.1.2

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

$\frac{- 5 \cos^{2} (x) + 5 \sin (x) + 5 (- \sin (x) \sin (x))}{2 \cos^{2} (x)}$

Step 11.3.1.3

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

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Step 11.3.1.3.1

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

$\frac{- 5 \cos^{2} (x) + 5 \sin (x) + 5 (- (\sin^{1} (x) \sin (x)))}{2 \cos^{2} (x)}$

Step 11.3.1.3.2

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

$\frac{- 5 \cos^{2} (x) + 5 \sin (x) + 5 (- (\sin^{1} (x) \sin^{1} (x)))}{2 \cos^{2} (x)}$

Step 11.3.1.3.3

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

$\frac{- 5 \cos^{2} (x) + 5 \sin (x) + 5 (- {\sin (x)}^{1 + 1})}{2 \cos^{2} (x)}$

Step 11.3.1.3.4

Add $1$ and $1$ .

$\frac{- 5 \cos^{2} (x) + 5 \sin (x) + 5 (- \sin^{2} (x))}{2 \cos^{2} (x)}$

Step 11.3.1.4

Multiply $- 1$ by $5$ .

$\frac{- 5 \cos^{2} (x) + 5 \sin (x) - 5 \sin^{2} (x)}{2 \cos^{2} (x)}$

Step 11.3.2

Move $- 5 \sin^{2} (x)$ .

$\frac{- 5 \cos^{2} (x) - 5 \sin^{2} (x) + 5 \sin (x)}{2 \cos^{2} (x)}$

Step 11.3.3

Factor $- 5$ out of $- 5 \cos^{2} (x)$ .

$\frac{- 5 (\cos^{2} (x)) - 5 \sin^{2} (x) + 5 \sin (x)}{2 \cos^{2} (x)}$

Step 11.3.4

Factor $- 5$ out of $- 5 \sin^{2} (x)$ .

$\frac{- 5 (\cos^{2} (x)) - 5 (\sin^{2} (x)) + 5 \sin (x)}{2 \cos^{2} (x)}$

Step 11.3.5

Factor $- 5$ out of $- 5 (\cos^{2} (x)) - 5 (\sin^{2} (x))$ .

$\frac{- 5 (\cos^{2} (x) + \sin^{2} (x)) + 5 \sin (x)}{2 \cos^{2} (x)}$

Step 11.3.6

Rearrange terms.

$\frac{- 5 (\sin^{2} (x) + \cos^{2} (x)) + 5 \sin (x)}{2 \cos^{2} (x)}$

Step 11.3.7

Apply pythagorean identity.

$\frac{- 5 \cdot 1 + 5 \sin (x)}{2 \cos^{2} (x)}$

Step 11.3.8

Multiply $- 5$ by $1$ .

$\frac{- 5 + 5 \sin (x)}{2 \cos^{2} (x)}$

Step 11.4

Factor $5$ out of $- 5 + 5 \sin (x)$ .

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Step 11.4.1

Factor $5$ out of $- 5$ .

$\frac{5 \cdot - 1 + 5 \sin (x)}{2 \cos^{2} (x)}$

Step 11.4.2

Factor $5$ out of $5 \cdot - 1 + 5 \sin (x)$ .

$\frac{5 (- 1 + \sin (x))}{2 \cos^{2} (x)}$

Step 11.5

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

$\frac{5 (- 1 (1) + \sin (x))}{2 \cos^{2} (x)}$

Step 11.6

Factor $- 1$ out of $\sin (x)$ .

$\frac{5 (- 1 (1) - 1 (- \sin (x)))}{2 \cos^{2} (x)}$

Step 11.7

Factor $- 1$ out of $- 1 (1) - 1 (- \sin (x))$ .

$\frac{5 (- 1 (1 - \sin (x)))}{2 \cos^{2} (x)}$

Step 11.8

Move the negative in front of the fraction.

$- \frac{5 (1 - \sin (x))}{2 \cos^{2} (x)}$