Calculus Examples

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

Step 1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{e^{x}}{3 x^{2}}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\frac{e^{x}}{x^{2}}]$ .

$\frac{1}{3} \frac{d}{d x} [\frac{e^{x}}{x^{2}}]$

Step 2

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

$\frac{1}{3} \cdot \frac{x^{2} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 3

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

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Step 3.1

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

$\frac{1}{3} \cdot \frac{x^{2} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 3.2

Multiply $2$ by $2$ .

$\frac{1}{3} \cdot \frac{x^{2} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4

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

$\frac{1}{3} \cdot \frac{x^{2} e^{x} - e^{x} \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

$\frac{1}{3} \cdot \frac{x^{2} e^{x} - e^{x} (2 x)}{x^{4}}$

Step 5.2

Combine fractions.

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Step 5.2.1

Multiply $2$ by $- 1$ .

$\frac{1}{3} \cdot \frac{x^{2} e^{x} - 2 e^{x} x}{x^{4}}$

Step 5.2.2

Multiply $\frac{1}{3}$ by $\frac{x^{2} e^{x} - 2 e^{x} x}{x^{4}}$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Reorder terms.

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

Step 6.3

Simplify the numerator.

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Step 6.3.1

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

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Step 6.3.1.1

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

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

Step 6.3.1.2

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

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

Step 6.3.1.3

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

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

Step 6.3.2

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

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Step 6.3.2.1

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

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

Step 6.3.2.2

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

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

Step 6.3.2.3

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

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

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $x$ out of $3 x^{4}$ .

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

Step 6.4.2.2

Cancel the common factor.

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

Step 6.4.2.3

Rewrite the expression.

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