Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \frac{- \cos (x)}{1 + \sin (x)}$ .

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Step 1.1

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

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 1.3

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

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

Step 2

Move the negative in front of the fraction.

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

Step 3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{\cos (x)}{1 + \sin (x)}$ .

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

Step 4

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 $g (x) = 1 + \sin (x)$ .

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

Step 5

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

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

Step 6

Differentiate.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Add $1$ and $1$ .

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

Step 12

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

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

Step 13

Simplify the expression.

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Step 13.1

Multiply $\frac{\cos (x)}{1 + \sin (x)}$ by $0$ .

$2 \frac{- \cos (x)}{1 + \sin (x)} (- \frac{(1 + \sin (x)) (- \sin (x)) - \cos^{2} (x)}{{(1 + \sin (x))}^{2}} + 0)$

Step 13.2

Add $- \frac{(1 + \sin (x)) (- \sin (x)) - \cos^{2} (x)}{{(1 + \sin (x))}^{2}}$ and $0$ .

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

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

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

Step 14.2

Simplify the numerator.

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Step 14.2.1

Simplify each term.

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Step 14.2.1.1

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

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

Step 14.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 14.2.1.3

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

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Step 14.2.1.3.1

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

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

Step 14.2.1.3.2

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

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

Step 14.2.1.3.3

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

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

Step 14.2.1.3.4

Add $1$ and $1$ .

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

Step 14.2.2

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

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

Step 14.2.3

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

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

Step 14.2.4

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

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

Step 14.2.5

Apply pythagorean identity.

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

Step 14.2.6

Multiply $- 1$ by $1$ .

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

Step 14.3

Combine terms.

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Step 14.3.1

Cancel the common factor of $- \sin (x) - 1$ and ${(1 + \sin (x))}^{2}$ .

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Step 14.3.1.1

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

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

Step 14.3.1.2

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

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

Step 14.3.1.3

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

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

Step 14.3.1.4

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

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

Step 14.3.1.5

Reorder terms.

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

Step 14.3.1.6

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

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

Step 14.3.1.7

Cancel the common factors.

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Step 14.3.1.7.1

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

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

Step 14.3.1.7.2

Cancel the common factor.

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

Step 14.3.1.7.3

Rewrite the expression.

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

Step 14.3.2

Move the negative in front of the fraction.

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

Step 14.3.3

Multiply $- 1$ by $- 1$ .

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

Step 14.3.4

Multiply $\frac{1}{1 + \sin (x)}$ by $1$ .

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

Step 15

Combine terms.

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Step 15.1

Move the negative in front of the fraction.

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

Step 15.2

Multiply $- 1$ by $2$ .

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

Step 15.3

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

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

Step 15.4

Move the negative in front of the fraction.

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

Step 15.5

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

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

Step 15.6

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

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

Step 15.7

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

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

Step 15.8

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

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

Step 15.9

Add $1$ and $1$ .

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