Calculus Examples

$f (x) = \frac{e^{x}}{x^{2}}$

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 $g (x) = x^{2}$ .

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

Step 2

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

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

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.1.1.2

Multiply $- 1$ by $2$ .

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

Step 4.1.2

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $2$ .

$\frac{x^{2} e^{x} - 2 x e^{x}}{x^{4}}$

Step 4.3

Reorder terms.

$\frac{e^{x} x^{2} - 2 e^{x} x}{x^{4}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

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

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Step 4.4.1.1

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

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

Step 4.4.1.2

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

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

Step 4.4.1.3

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

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

Step 4.4.2

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

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Step 4.4.2.1

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

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

Step 4.4.2.2

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

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

Step 4.4.2.3

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

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

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

Step 4.5

Cancel the common factor of $x$ and $x^{4}$ .

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Step 4.5.1

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

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

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $x$ out of $x^{4}$ .

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

Step 4.5.2.2

Cancel the common factor.

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

Step 4.5.2.3

Rewrite the expression.

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