Calculus Examples

$- \frac{\sqrt[5]{x}}{2}$

Step 1

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

$\frac{d}{d x} [- \frac{x^{\frac{1}{5}}}{2}]$

Step 2

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{x^{\frac{1}{5}}}{2}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x^{\frac{1}{5}}]$ .

$- \frac{1}{2} \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 3

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

$- \frac{1}{2} (\frac{1}{5} x^{\frac{1}{5} - 1})$

Step 4

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

$- \frac{1}{2} (\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}})$

Step 5

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

$- \frac{1}{2} (\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}})$

Step 6

Combine the numerators over the common denominator.

$- \frac{1}{2} (\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}})$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $5$ .

$- \frac{1}{2} (\frac{1}{5} x^{\frac{1 - 5}{5}})$

Step 7.2

Subtract $5$ from $1$ .

$- \frac{1}{2} (\frac{1}{5} x^{\frac{- 4}{5}})$

Step 8

Move the negative in front of the fraction.

$- \frac{1}{2} (\frac{1}{5} x^{- \frac{4}{5}})$

Step 9

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

$- \frac{1}{2} \cdot \frac{x^{- \frac{4}{5}}}{5}$

Step 10

Multiply $\frac{x^{- \frac{4}{5}}}{5}$ by $\frac{1}{2}$ .

$- \frac{x^{- \frac{4}{5}}}{5 \cdot 2}$

Step 11

Simplify the expression.

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Step 11.1

Multiply $5$ by $2$ .

$- \frac{x^{- \frac{4}{5}}}{10}$

Step 11.2

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

$- \frac{1}{10 x^{\frac{4}{5}}}$