Calculus Examples

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

Step 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 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

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 4

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

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

Step 5

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 6

Multiply.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2

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

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

Step 7

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 8

Simplify.

Tap for more steps...

Step 8.1

Apply the distributive property.

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

Step 8.2

Apply the distributive property.

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

Step 8.3

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 8.4

Simplify the numerator.

Tap for more steps...

Step 8.4.1

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

Tap for more steps...

Step 8.4.1.1

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

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

Step 8.4.1.2

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

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

Step 8.4.2

Simplify each term.

Tap for more steps...

Step 8.4.2.1

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

Tap for more steps...

Step 8.4.2.1.1

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

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

Step 8.4.2.1.2

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

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

Step 8.4.2.1.3

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 8.4.2.1.4

Add $1$ and $1$ .

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

Step 8.4.2.2

Multiply $- - \cos (q)$ .

Tap for more steps...

Step 8.4.2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 8.4.2.2.2

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

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

Step 8.4.2.3

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

Tap for more steps...

Step 8.4.2.3.1

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

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

Step 8.4.2.3.2

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

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

Step 8.4.2.3.3

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 8.4.2.3.4

Add $1$ and $1$ .

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

Step 8.4.3

Apply pythagorean identity.

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

Step 8.5

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

$\csc^{2} (q)$