Calculus Examples

$p = \frac{\sec (q) + \csc (q)}{\csc (q)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d q} (p) = \frac{d}{d q} (\frac{\sec (q) + \csc (q)}{\csc (q)})$

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 $f (q) = \sec (q) + \csc (q)$ and $g (q) = \csc (q)$ .

$\frac{\csc (q) \frac{d}{d q} [\sec (q) + \csc (q)] - (\sec (q) + \csc (q)) \frac{d}{d q} [\csc (q)]}{\csc^{2} (q)}$

Step 3.2

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

$\frac{\csc (q) (\frac{d}{d q} [\sec (q)] + \frac{d}{d q} [\csc (q)]) - (\sec (q) + \csc (q)) \frac{d}{d q} [\csc (q)]}{\csc^{2} (q)}$

Step 3.3

The derivative of $\sec (q)$ with respect to $q$ is $\sec (q) \tan (q)$ .

$\frac{\csc (q) (\sec (q) \tan (q) + \frac{d}{d q} [\csc (q)]) - (\sec (q) + \csc (q)) \frac{d}{d q} [\csc (q)]}{\csc^{2} (q)}$

Step 3.4

The derivative of $\csc (q)$ with respect to $q$ is $- \csc (q) \cot (q)$ .

$\frac{\csc (q) (\sec (q) \tan (q) - \csc (q) \cot (q)) - (\sec (q) + \csc (q)) \frac{d}{d q} [\csc (q)]}{\csc^{2} (q)}$

Step 3.5

The derivative of $\csc (q)$ with respect to $q$ is $- \csc (q) \cot (q)$ .

$\frac{\csc (q) (\sec (q) \tan (q) - \csc (q) \cot (q)) - (\sec (q) + \csc (q)) (- \csc (q) \cot (q))}{\csc^{2} (q)}$

Step 3.6

Simplify with factoring out.

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Step 3.6.1

Multiply $- 1$ by $- 1$ .

$\frac{\csc (q) (\sec (q) \tan (q) - \csc (q) \cot (q)) + 1 (\sec (q) + \csc (q)) (\csc (q) \cot (q))}{\csc^{2} (q)}$

Step 3.6.2

Multiply $\sec (q) + \csc (q)$ by $1$ .

$\frac{\csc (q) (\sec (q) \tan (q) - \csc (q) \cot (q)) + (\sec (q) + \csc (q)) (\csc (q) \cot (q))}{\csc^{2} (q)}$

Step 3.6.3

Factor $\csc (q)$ out of $\csc (q) (\sec (q) \tan (q) - \csc (q) \cot (q)) + (\sec (q) + \csc (q)) \csc (q) \cot (q)$ .

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Step 3.6.3.1

Factor $\csc (q)$ out of $(\sec (q) + \csc (q)) \csc (q) \cot (q)$ .

$\frac{\csc (q) (\sec (q) \tan (q) - \csc (q) \cot (q)) + \csc (q) ((\sec (q) + \csc (q)) \cot (q))}{\csc^{2} (q)}$

Step 3.6.3.2

Factor $\csc (q)$ out of $\csc (q) (\sec (q) \tan (q) - \csc (q) \cot (q)) + \csc (q) ((\sec (q) + \csc (q)) \cot (q))$ .

$\frac{\csc (q) (\sec (q) \tan (q) - \csc (q) \cot (q) + (\sec (q) + \csc (q)) \cot (q))}{\csc^{2} (q)}$

Step 3.7

Cancel the common factors.

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Step 3.7.1

Factor $\csc (q)$ out of $\csc^{2} (q)$ .

$\frac{\csc (q) (\sec (q) \tan (q) - \csc (q) \cot (q) + (\sec (q) + \csc (q)) \cot (q))}{\csc (q) \csc (q)}$

Step 3.7.2

Cancel the common factor.

$\frac{\csc (q) (\sec (q) \tan (q) - \csc (q) \cot (q) + (\sec (q) + \csc (q)) \cot (q))}{\csc (q) \csc (q)}$

Step 3.7.3

Rewrite the expression.

$\frac{\sec (q) \tan (q) - \csc (q) \cot (q) + (\sec (q) + \csc (q)) \cot (q)}{\csc (q)}$

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

$\frac{\sec (q) \tan (q) - \csc (q) \cot (q) + \sec (q) \cot (q) + \csc (q) \cot (q)}{\csc (q)}$

Step 3.8.2

Simplify the numerator.

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Step 3.8.2.1

Combine the opposite terms in $\sec (q) \tan (q) - \csc (q) \cot (q) + \sec (q) \cot (q) + \csc (q) \cot (q)$ .

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Step 3.8.2.1.1

Add $- \csc (q) \cot (q)$ and $\csc (q) \cot (q)$ .

$\frac{\sec (q) \tan (q) + \sec (q) \cot (q) + 0}{\csc (q)}$

Step 3.8.2.1.2

Add $\sec (q) \tan (q) + \sec (q) \cot (q)$ and $0$ .

$\frac{\sec (q) \tan (q) + \sec (q) \cot (q)}{\csc (q)}$

Step 3.8.2.2

Simplify each term.

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Step 3.8.2.2.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.8.2.2.1.1

Reorder $\sec (q)$ and $\cot (q)$ .

$\frac{\sec (q) \tan (q) + \cot (q) \sec (q)}{\csc (q)}$

Step 3.8.2.2.1.2

Rewrite $\sec (q) \cot (q)$ in terms of sines and cosines.

$\frac{\sec (q) \tan (q) + \frac{\cos (q)}{\sin (q)} \cdot \frac{1}{\cos (q)}}{\csc (q)}$

Step 3.8.2.2.1.3

Cancel the common factors.

$\frac{\sec (q) \tan (q) + \frac{1}{\sin (q)}}{\csc (q)}$

Step 3.8.2.2.2

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

$\frac{\sec (q) \tan (q) + \csc (q)}{\csc (q)}$

Step 4

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

$p' = \frac{\sec (q) \tan (q) + \csc (q)}{\csc (q)}$

Step 5

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

$\frac{d p}{d q} = \frac{\sec (q) \tan (q) + \csc (q)}{\csc (q)}$