Calculus Examples

$y = \sqrt[5]{z}$

Step 1

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

$\frac{d}{d z} [z^{\frac{1}{5}}]$

Step 2

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

$\frac{1}{5} z^{\frac{1}{5} - 1}$

Step 3

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

$\frac{1}{5} z^{\frac{1}{5} - 1 \cdot \frac{5}{5}}$

Step 4

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

$\frac{1}{5} z^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}}$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{5} z^{\frac{1 - 1 \cdot 5}{5}}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $5$ .

$\frac{1}{5} z^{\frac{1 - 5}{5}}$

Step 6.2

Subtract $5$ from $1$ .

$\frac{1}{5} z^{\frac{- 4}{5}}$

Step 7

Move the negative in front of the fraction.

$\frac{1}{5} z^{- \frac{4}{5}}$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{5} \cdot \frac{1}{z^{\frac{4}{5}}}$

Step 8.2

Multiply $\frac{1}{5}$ by $\frac{1}{z^{\frac{4}{5}}}$ .

$\frac{1}{5 z^{\frac{4}{5}}}$