Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Multiply.

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Step 3.7.1

Multiply $- 1$ by $- 1$ .

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

Step 3.7.2

Multiply $\cos (q) - \sin (q)$ by $1$ .

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Apply the distributive property.

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

Step 3.8.3

Simplify the numerator.

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Step 3.8.3.1

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

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Step 3.8.3.1.1

Reorder the factors in the terms $\cos (q) (- \sin (q))$ and $\cos (q) \sin (q)$ .

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

Step 3.8.3.1.2

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

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

Step 3.8.3.1.3

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

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

Step 3.8.3.2

Simplify each term.

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Step 3.8.3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.8.3.2.2

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

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Step 3.8.3.2.2.1

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

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

Step 3.8.3.2.2.2

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

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

Step 3.8.3.2.2.3

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

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

Step 3.8.3.2.2.4

Add $1$ and $1$ .

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

Step 3.8.3.2.3

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

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Step 3.8.3.2.3.1

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

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

Step 3.8.3.2.3.2

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

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

Step 3.8.3.2.3.3

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

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

Step 3.8.3.2.3.4

Add $1$ and $1$ .

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

Step 3.8.3.3

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

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

Step 3.8.3.4

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

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

Step 3.8.3.5

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

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

Step 3.8.3.6

Rearrange terms.

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

Step 3.8.3.7

Apply pythagorean identity.

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

Step 3.8.3.8

Multiply $- 1$ by $1$ .

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

Step 3.8.4

Combine terms.

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Step 3.8.4.1

Move the negative in front of the fraction.

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

Step 3.8.4.2

Convert from $\frac{1}{\cos^{2} (q)}$ to $\sec^{2} (q)$ .

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

Step 4

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

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

Step 5

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

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