Calculus Examples

$\frac{e^{x}}{\cos (x)}$

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 $f (x) = e^{x}$ and $g (x) = \cos (x)$ .

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

Step 2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 3

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

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

Step 4

Multiply.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Multiply $e^{x}$ by $1$ .

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

Step 5

Simplify.

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Step 5.1

Reorder factors in $\cos (x) e^{x} + e^{x} \sin (x)$ .

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

Step 5.2

Factor $e^{x}$ out of $e^{x} \cos (x) + e^{x} \sin (x)$ .

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Step 5.2.1

Factor $e^{x}$ out of $e^{x} \cos (x)$ .

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

Step 5.2.2

Factor $e^{x}$ out of $e^{x} \sin (x)$ .

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

Step 5.2.3

Factor $e^{x}$ out of $e^{x} (\cos (x)) + e^{x} (\sin (x))$ .

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