Calculus Examples

$h (x) = {(\frac{\cos (x)}{1 + \sin (x)})}^{5}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = \frac{\cos (x)}{1 + \sin (x)}$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $\frac{\cos (x)}{1 + \sin (x)}$ .

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$5 u^{4} \frac{d}{d x} [\frac{\cos (x)}{1 + \sin (x)}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{\cos (x)}{1 + \sin (x)}$ .

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{d}{d x} [\frac{\cos (x)}{1 + \sin (x)}]$

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

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{(1 + \sin (x)) \frac{d}{d x} [\cos (x)] - \cos (x) \frac{d}{d x} [1 + \sin (x)]}{{(1 + \sin (x))}^{2}}$

Step 3

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

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{(1 + \sin (x)) (- \sin (x)) - \cos (x) \frac{d}{d x} [1 + \sin (x)]}{{(1 + \sin (x))}^{2}}$

Step 4

Differentiate.

Tap for more steps...

Step 4.1

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

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{(1 + \sin (x)) (- \sin (x)) - \cos (x) (\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)])}{{(1 + \sin (x))}^{2}}$

Step 4.2

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

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{(1 + \sin (x)) (- \sin (x)) - \cos (x) (0 + \frac{d}{d x} [\sin (x)])}{{(1 + \sin (x))}^{2}}$

Step 4.3

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

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{(1 + \sin (x)) (- \sin (x)) - \cos (x) \frac{d}{d x} [\sin (x)]}{{(1 + \sin (x))}^{2}}$

Step 5

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

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{(1 + \sin (x)) (- \sin (x)) - \cos (x) \cos (x)}{{(1 + \sin (x))}^{2}}$

Step 6

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

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{(1 + \sin (x)) (- \sin (x)) - (\cos^{1} (x) \cos (x))}{{(1 + \sin (x))}^{2}}$

Step 7

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

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{(1 + \sin (x)) (- \sin (x)) - (\cos^{1} (x) \cos^{1} (x))}{{(1 + \sin (x))}^{2}}$

Step 8

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

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{(1 + \sin (x)) (- \sin (x)) - {\cos (x)}^{1 + 1}}{{(1 + \sin (x))}^{2}}$

Step 9

Add $1$ and $1$ .

$5 {(\frac{\cos (x)}{1 + \sin (x)})}^{4} \frac{(1 + \sin (x)) (- \sin (x)) - \cos^{2} (x)}{{(1 + \sin (x))}^{2}}$

Step 10

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

$\frac{((1 + \sin (x)) (- \sin (x)) - \cos^{2} (x)) \cdot 5}{{(1 + \sin (x))}^{2}} {(\frac{\cos (x)}{1 + \sin (x)})}^{4}$

Step 11

Move $5$ to the left of $(1 + \sin (x)) (- \sin (x)) - \cos^{2} (x)$ .

$\frac{5 \cdot ((1 + \sin (x)) (- \sin (x)) - \cos^{2} (x))}{{(1 + \sin (x))}^{2}} {(\frac{\cos (x)}{1 + \sin (x)})}^{4}$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Apply the product rule to $\frac{\cos (x)}{1 + \sin (x)}$ .

$\frac{5 ((1 + \sin (x)) (- \sin (x)) - \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.2

Apply the distributive property.

$\frac{5 (1 (- \sin (x)) + \sin (x) (- \sin (x)) - \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.3

Apply the distributive property.

$\frac{5 (1 (- \sin (x))) + 5 (\sin (x) (- \sin (x))) + 5 (- \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.4

Combine terms.

Tap for more steps...

Step 12.4.1

Multiply $- 1$ by $1$ .

$\frac{5 (- 1 \sin (x)) + 5 (\sin (x) (- \sin (x))) + 5 (- \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.4.2

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

$\frac{5 (- \sin (x)) + 5 (\sin (x) (- \sin (x))) + 5 (- \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.4.3

Multiply $- 1$ by $5$ .

$\frac{- 5 \sin (x) + 5 (\sin (x) (- \sin (x))) + 5 (- \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.4.4

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

$\frac{- 5 \sin (x) + 5 (- (\sin^{1} (x) \sin (x))) + 5 (- \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.4.5

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

$\frac{- 5 \sin (x) + 5 (- (\sin^{1} (x) \sin^{1} (x))) + 5 (- \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.4.6

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

$\frac{- 5 \sin (x) + 5 (- {\sin (x)}^{1 + 1}) + 5 (- \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.4.7

Add $1$ and $1$ .

$\frac{- 5 \sin (x) + 5 (- \sin^{2} (x)) + 5 (- \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.4.8

Multiply $- 1$ by $5$ .

$\frac{- 5 \sin (x) - 5 \sin^{2} (x) + 5 (- \cos^{2} (x))}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.4.9

Multiply $- 1$ by $5$ .

$\frac{- 5 \sin (x) - 5 \sin^{2} (x) - 5 \cos^{2} (x)}{{(1 + \sin (x))}^{2}} \cdot \frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$

Step 12.4.10

Multiply $\frac{- 5 \sin (x) - 5 \sin^{2} (x) - 5 \cos^{2} (x)}{{(1 + \sin (x))}^{2}}$ by $\frac{\cos^{4} (x)}{{(1 + \sin (x))}^{4}}$ .

$\frac{(- 5 \sin (x) - 5 \sin^{2} (x) - 5 \cos^{2} (x)) \cos^{4} (x)}{{(1 + \sin (x))}^{2} {(1 + \sin (x))}^{4}}$

Step 12.4.11

Multiply ${(1 + \sin (x))}^{2}$ by ${(1 + \sin (x))}^{4}$ by adding the exponents.

Tap for more steps...

Step 12.4.11.1

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

$\frac{(- 5 \sin (x) - 5 \sin^{2} (x) - 5 \cos^{2} (x)) \cos^{4} (x)}{{(1 + \sin (x))}^{2 + 4}}$

Step 12.4.11.2

Add $2$ and $4$ .

$\frac{(- 5 \sin (x) - 5 \sin^{2} (x) - 5 \cos^{2} (x)) \cos^{4} (x)}{{(1 + \sin (x))}^{6}}$

Step 12.5

Reorder terms.

$\frac{\cos^{4} (x) (- 5 \sin (x) - 5 \sin^{2} (x) - 5 \cos^{2} (x))}{{(1 + \sin (x))}^{6}}$

Step 12.6

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

$\frac{\cos^{4} (x) (- 5 \sin (x) - 5 (\sin^{2} (x)) - 5 \cos^{2} (x))}{{(1 + \sin (x))}^{6}}$

Step 12.7

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

$\frac{\cos^{4} (x) (- 5 \sin (x) - 5 (\sin^{2} (x)) - 5 (\cos^{2} (x)))}{{(1 + \sin (x))}^{6}}$

Step 12.8

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

$\frac{\cos^{4} (x) (- 5 \sin (x) - 5 (\sin^{2} (x) + \cos^{2} (x)))}{{(1 + \sin (x))}^{6}}$

Step 12.9

Apply pythagorean identity.

$\frac{\cos^{4} (x) (- 5 \sin (x) - 5 \cdot 1)}{{(1 + \sin (x))}^{6}}$

Step 12.10

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

Tap for more steps...

Step 12.10.1

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

$\frac{\cos^{4} (x) (- 5 (\sin (x)) - 5 \cdot 1)}{{(1 + \sin (x))}^{6}}$

Step 12.10.2

Factor $- 5$ out of $- 5 \cdot 1$ .

$\frac{\cos^{4} (x) (- 5 (\sin (x)) - 5 (1))}{{(1 + \sin (x))}^{6}}$

Step 12.10.3

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

$\frac{\cos^{4} (x) (- 5 (\sin (x) + 1))}{{(1 + \sin (x))}^{6}}$

$\frac{\cos^{4} (x) \cdot - 5 (\sin (x) + 1)}{{(1 + \sin (x))}^{6}}$

Step 12.11

Cancel the common factor of $\sin (x) + 1$ and ${(1 + \sin (x))}^{6}$ .

Tap for more steps...

Step 12.11.1

Reorder terms.

$\frac{\cos^{4} (x) \cdot - 5 (1 + \sin (x))}{{(1 + \sin (x))}^{6}}$

Step 12.11.2

Factor $1 + \sin (x)$ out of $\cos^{4} (x) \cdot - 5 (1 + \sin (x))$ .

$\frac{(1 + \sin (x)) (\cos^{4} (x) \cdot - 5)}{{(1 + \sin (x))}^{6}}$

Step 12.11.3

Cancel the common factors.

Tap for more steps...

Step 12.11.3.1

Factor $1 + \sin (x)$ out of ${(1 + \sin (x))}^{6}$ .

$\frac{(1 + \sin (x)) (\cos^{4} (x) \cdot - 5)}{(1 + \sin (x)) {(1 + \sin (x))}^{5}}$

Step 12.11.3.2

Cancel the common factor.

$\frac{(1 + \sin (x)) (\cos^{4} (x) \cdot - 5)}{(1 + \sin (x)) {(1 + \sin (x))}^{5}}$

Step 12.11.3.3

Rewrite the expression.

$\frac{\cos^{4} (x) \cdot - 5}{{(1 + \sin (x))}^{5}}$

Step 12.12

Move $- 5$ to the left of $\cos^{4} (x)$ .

$\frac{- 5 \cdot \cos^{4} (x)}{{(1 + \sin (x))}^{5}}$

Step 12.13

Move the negative in front of the fraction.

$- \frac{5 \cos^{4} (x)}{{(1 + \sin (x))}^{5}}$