Calculus Examples

$\frac{e^{x}}{x}$

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 $g (x) = x$ .

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

Step 2

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{x e^{x} - e^{x} \cdot 1}{x^{2}}$

Step 3.2

Multiply $- 1$ by $1$ .

$\frac{x e^{x} - e^{x}}{x^{2}}$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$\frac{e^{x} x - e^{x}}{x^{2}}$

Step 4.2

Factor $e^{x}$ out of $e^{x} x - e^{x}$ .

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Step 4.2.1

Factor $e^{x}$ out of $e^{x} x$ .

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

Step 4.2.2

Factor $e^{x}$ out of $- e^{x}$ .

$\frac{e^{x} (x) + e^{x} \cdot - 1}{x^{2}}$

Step 4.2.3

Factor $e^{x}$ out of $e^{x} (x) + e^{x} \cdot - 1$ .

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