Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

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

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Multiply.

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Step 3.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.5.2

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Add $1$ and $1$ .

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

Step 3.9

Simplify.

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Step 3.9.1

Apply the distributive property.

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

Step 3.9.2

Simplify the numerator.

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Step 3.9.2.1

Simplify each term.

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Step 3.9.2.1.1

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

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

Step 3.9.2.1.2

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

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Step 3.9.2.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.9.2.1.2.2

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

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

Step 3.9.2.1.2.3

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

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

Step 3.9.2.1.2.4

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

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

Step 3.9.2.1.2.5

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

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

Step 3.9.2.1.2.6

Add $1$ and $1$ .

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

Step 3.9.2.2

Apply pythagorean identity.

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

Step 3.9.3

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

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Step 3.9.3.1

Reorder terms.

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

Step 3.9.3.2

Multiply by $1$ .

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

Step 3.9.3.3

Cancel the common factors.

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Step 3.9.3.3.1

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

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

Step 3.9.3.3.2

Cancel the common factor.

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

Step 3.9.3.3.3

Rewrite the expression.

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

Step 4

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

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

Step 5

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

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