Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply.

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Step 3.6.1

Multiply $- 1$ by $- 1$ .

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

Step 3.6.2

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

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

Step 3.7

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

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Apply the distributive property.

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

Step 3.8.3

Apply the distributive property.

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

Step 3.8.4

Simplify the numerator.

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Step 3.8.4.1

Combine the opposite terms in $\sin (x) \cos (x) + \sin (x) \sin (x) - \sin (x) \cos (x) - - \cos (x) \cos (x)$ .

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Step 3.8.4.1.1

Subtract $\sin (x) \cos (x)$ from $\sin (x) \cos (x)$ .

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

Step 3.8.4.1.2

Add $\sin (x) \sin (x)$ and $0$ .

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

Step 3.8.4.2

Simplify each term.

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Step 3.8.4.2.1

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

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Step 3.8.4.2.1.1

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

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

Step 3.8.4.2.1.2

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

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

Step 3.8.4.2.1.3

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

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

Step 3.8.4.2.1.4

Add $1$ and $1$ .

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

Step 3.8.4.2.2

Multiply $- - \cos (x)$ .

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Step 3.8.4.2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.8.4.2.2.2

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

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

Step 3.8.4.2.3

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

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Step 3.8.4.2.3.1

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

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

Step 3.8.4.2.3.2

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

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

Step 3.8.4.2.3.3

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

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

Step 3.8.4.2.3.4

Add $1$ and $1$ .

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

Step 3.8.4.3

Apply pythagorean identity.

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

Step 3.8.5

Convert from $\frac{1}{\sin^{2} (x)}$ to $\csc^{2} (x)$ .

$\csc^{2} (x)$

Step 4

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

$y' = \csc^{2} (x)$

Step 5

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

$\frac{d y}{d x} = \csc^{2} (x)$