Calculus Examples

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

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 $f (x) = \sin (x)$ and $g (x) = \cos (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Simplify the numerator.

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Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rearrange terms.

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

Apply pythagorean identity.

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

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

$\sec^{2} (x)$