Calculus Examples

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

Step 1

This derivative could not be completed using the product rule. Mathway will use another method.

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Move $2$ to the left of $n$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Reorder factors in $x^{n} e^{x} - e^{x} (n x^{n - 1})$ .

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

Step 6.2

Reorder terms.

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

Step 6.3

Factor $e^{x}$ out of $e^{x} x^{n} - e^{x} x^{n - 1} n$ .

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Step 6.3.1

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

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

Step 6.3.2

Factor $e^{x}$ out of $- e^{x} x^{n - 1} n$ .

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

Step 6.3.3

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

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

Step 6.4

Reorder factors in $\frac{e^{x} (x^{n} - x^{n - 1} n)}{x^{2 n}}$ .

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