Calculus Examples

$e^{\sqrt[5]{x}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$\frac{d}{d x} [e^{x^{\frac{1}{5}}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{\frac{1}{5}}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{5}}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 2.2

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

$e^{u} \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 2.3

Replace all occurrences of $u$ with $x^{\frac{1}{5}}$ .

$e^{x^{\frac{1}{5}}} \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{5}$ .

$e^{x^{\frac{1}{5}}} (\frac{1}{5} x^{\frac{1}{5} - 1})$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$e^{x^{\frac{1}{5}}} (\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}})$

Step 5

Combine $- 1$ and $\frac{5}{5}$ .

$e^{x^{\frac{1}{5}}} (\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}})$

Step 6

Combine the numerators over the common denominator.

$e^{x^{\frac{1}{5}}} (\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}})$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $5$ .

$e^{x^{\frac{1}{5}}} (\frac{1}{5} x^{\frac{1 - 5}{5}})$

Step 7.2

Subtract $5$ from $1$ .

$e^{x^{\frac{1}{5}}} (\frac{1}{5} x^{\frac{- 4}{5}})$

Step 8

Move the negative in front of the fraction.

$e^{x^{\frac{1}{5}}} (\frac{1}{5} x^{- \frac{4}{5}})$

Step 9

Combine $\frac{1}{5}$ and $x^{- \frac{4}{5}}$ .

$e^{x^{\frac{1}{5}}} \frac{x^{- \frac{4}{5}}}{5}$

Step 10

Combine $e^{x^{\frac{1}{5}}}$ and $\frac{x^{- \frac{4}{5}}}{5}$ .

$\frac{e^{x^{\frac{1}{5}}} x^{- \frac{4}{5}}}{5}$

Step 11

Move $x^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{e^{x^{\frac{1}{5}}}}{5 x^{\frac{4}{5}}}$