Calculus Examples

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

Step 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 $g (x) = \cos (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Multiply.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2

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

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

Step 7

Simplify.

Tap for more steps...

Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Simplify the numerator.

Tap for more steps...

Step 7.3.1

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

Tap for more steps...

Step 7.3.1.1

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

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

Step 7.3.1.2

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

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

Step 7.3.1.3

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

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

Step 7.3.2

Simplify each term.

Tap for more steps...

Step 7.3.2.1

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

Tap for more steps...

Step 7.3.2.1.1

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

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

Step 7.3.2.1.2

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

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

Step 7.3.2.1.3

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

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

Step 7.3.2.1.4

Add $1$ and $1$ .

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

Step 7.3.2.2

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

Tap for more steps...

Step 7.3.2.2.1

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

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

Step 7.3.2.2.2

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

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

Step 7.3.2.2.3

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

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

Step 7.3.2.2.4

Add $1$ and $1$ .

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

Step 7.3.3

Rearrange terms.

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

Step 7.3.4

Apply pythagorean identity.

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

Step 7.4

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

$\sec^{2} (x)$