Calculus Examples

$p = \frac{\sin (q) + \cos (q)}{\sin (q)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d q} (p) = \frac{d}{d q} (\frac{\sin (q) + \cos (q)}{\sin (q)})$

Step 2

The derivative of $p$ with respect to $q$ is $p'$ .

$p'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d q} [\frac{f (q)}{g (q)}]$ is $\frac{g (q) \frac{d}{d q} [f (q)] - f (q) \frac{d}{d q} [g (q)]}{{g (q)}^{2}}$ where $f (q) = \sin (q) + \cos (q)$ and $g (q) = \sin (q)$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

Apply the distributive property.

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

Step 3.6.3

Apply the distributive property.

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

Step 3.6.4

Simplify the numerator.

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Step 3.6.4.1

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

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Step 3.6.4.1.1

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

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

Step 3.6.4.1.2

Add $\sin (q) (- \sin (q))$ and $0$ .

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

Step 3.6.4.2

Simplify each term.

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Step 3.6.4.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.6.4.2.2

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

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Step 3.6.4.2.2.1

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

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

Step 3.6.4.2.2.2

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

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

Step 3.6.4.2.2.3

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

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

Step 3.6.4.2.2.4

Add $1$ and $1$ .

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

Step 3.6.4.2.3

Multiply $- \cos (q) \cos (q)$ .

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Step 3.6.4.2.3.1

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

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

Step 3.6.4.2.3.2

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

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

Step 3.6.4.2.3.3

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

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

Step 3.6.4.2.3.4

Add $1$ and $1$ .

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

Step 3.6.4.3

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

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

Step 3.6.4.4

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

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

Step 3.6.4.5

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

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

Step 3.6.4.6

Apply pythagorean identity.

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

Step 3.6.4.7

Multiply $- 1$ by $1$ .

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

Step 3.6.5

Combine terms.

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Step 3.6.5.1

Move the negative in front of the fraction.

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

Step 3.6.5.2

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

$- \csc^{2} (q)$

Step 4

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

$p' = - \csc^{2} (q)$

Step 5

Replace $p'$ with $\frac{d p}{d q}$ .

$\frac{d p}{d q} = - \csc^{2} (q)$