Calculus Examples

$\frac{e^{x}}{x^{3}}$

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 $g (x) = x^{3}$ .

$\frac{x^{3} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{3}]}{{(x^{3})}^{2}}$

Step 2

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{3} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 2.2

Multiply $3$ by $2$ .

$\frac{x^{3} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3

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

$\frac{x^{3} e^{x} - e^{x} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{x^{3} e^{x} - e^{x} (3 x^{2})}{x^{6}}$

Step 4.2

Multiply $3$ by $- 1$ .

$\frac{x^{3} e^{x} - 3 e^{x} x^{2}}{x^{6}}$

Step 5

Simplify.

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Step 5.1

Reorder factors in $x^{3} e^{x} - 3 e^{x} x^{2}$ .

$\frac{x^{3} e^{x} - 3 x^{2} e^{x}}{x^{6}}$

Step 5.2

Reorder terms.

$\frac{e^{x} x^{3} - 3 e^{x} x^{2}}{x^{6}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Factor $x^{2}$ out of $e^{x} x^{3} - 3 e^{x} x^{2}$ .

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Step 5.3.1.1

Factor $x^{2}$ out of $e^{x} x^{3}$ .

$\frac{x^{2} (e^{x} x) - 3 e^{x} x^{2}}{x^{6}}$

Step 5.3.1.2

Factor $x^{2}$ out of $- 3 e^{x} x^{2}$ .

$\frac{x^{2} (e^{x} x) + x^{2} (- 3 e^{x})}{x^{6}}$

Step 5.3.1.3

Factor $x^{2}$ out of $x^{2} (e^{x} x) + x^{2} (- 3 e^{x})$ .

$\frac{x^{2} (e^{x} x - 3 e^{x})}{x^{6}}$

Step 5.3.2

Factor $e^{x}$ out of $e^{x} x - 3 e^{x}$ .

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Step 5.3.2.1

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

$\frac{x^{2} (e^{x} (x) - 3 e^{x})}{x^{6}}$

Step 5.3.2.2

Factor $e^{x}$ out of $- 3 e^{x}$ .

$\frac{x^{2} (e^{x} (x) + e^{x} \cdot - 3)}{x^{6}}$

Step 5.3.2.3

Factor $e^{x}$ out of $e^{x} (x) + e^{x} \cdot - 3$ .

$\frac{x^{2} (e^{x} (x - 3))}{x^{6}}$

$\frac{x^{2} e^{x} (x - 3)}{x^{6}}$

Step 5.4

Cancel the common factor of $x^{2}$ and $x^{6}$ .

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Step 5.4.1

Factor $x^{2}$ out of $x^{2} e^{x} (x - 3)$ .

$\frac{x^{2} (e^{x} (x - 3))}{x^{6}}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $x^{2}$ out of $x^{6}$ .

$\frac{x^{2} (e^{x} (x - 3))}{x^{2} x^{4}}$

Step 5.4.2.2

Cancel the common factor.

$\frac{x^{2} (e^{x} (x - 3))}{x^{2} x^{4}}$

Step 5.4.2.3

Rewrite the expression.

$\frac{e^{x} (x - 3)}{x^{4}}$