Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{2 \cos (x)}{1 + \sin (x)})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Add $1$ and $1$ .

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

Step 3.10

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

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

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

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

Step 3.11.2

Apply the distributive property.

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

Step 3.11.3

Simplify the numerator.

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Step 3.11.3.1

Simplify each term.

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Step 3.11.3.1.1

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

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

Step 3.11.3.1.2

Multiply $- 1$ by $2$ .

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

Step 3.11.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.11.3.1.4

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

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Step 3.11.3.1.4.1

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

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

Step 3.11.3.1.4.2

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

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

Step 3.11.3.1.4.3

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

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

Step 3.11.3.1.4.4

Add $1$ and $1$ .

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

Step 3.11.3.1.5

Multiply $- 1$ by $2$ .

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

Step 3.11.3.1.6

Multiply $- 1$ by $2$ .

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

Step 3.11.3.2

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

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

Step 3.11.3.3

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

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

Step 3.11.3.4

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

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

Step 3.11.3.5

Apply pythagorean identity.

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

Step 3.11.3.6

Multiply $- 2$ by $1$ .

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

Step 3.11.4

Reorder terms.

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

Step 3.11.5

Factor $- 2$ out of $- 2 - 2 \sin (x)$ .

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Step 3.11.5.1

Factor $- 2$ out of $- 2$ .

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

Step 3.11.5.2

Factor $- 2$ out of $- 2 \sin (x)$ .

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

Step 3.11.5.3

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

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

Step 3.11.6

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

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Step 3.11.6.1

Factor $1 + \sin (x)$ out of $- 2 (1 + \sin (x))$ .

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

Step 3.11.6.2

Cancel the common factors.

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Step 3.11.6.2.1

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

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

Step 3.11.6.2.2

Cancel the common factor.

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

Step 3.11.6.2.3

Rewrite the expression.

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

Step 3.11.7

Move the negative in front of the fraction.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{2}{1 + \sin (x)}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{2}{1 + \sin (x)}$