Calculus Examples

$\sqrt[8]{x^{7}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{x^{7}}$ as $x^{\frac{7}{8}}$ .

$\frac{d}{d x} [x^{\frac{7}{8}}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{8}$ .

$\frac{7}{8} x^{\frac{7}{8} - 1}$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{7}{8} x^{\frac{7}{8} - 1 \cdot \frac{8}{8}}$

Step 4

Combine $- 1$ and $\frac{8}{8}$ .

$\frac{7}{8} x^{\frac{7}{8} + \frac{- 1 \cdot 8}{8}}$

Step 5

Combine the numerators over the common denominator.

$\frac{7}{8} x^{\frac{7 - 1 \cdot 8}{8}}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $8$ .

$\frac{7}{8} x^{\frac{7 - 8}{8}}$

Step 6.2

Subtract $8$ from $7$ .

$\frac{7}{8} x^{\frac{- 1}{8}}$

Step 7

Move the negative in front of the fraction.

$\frac{7}{8} x^{- \frac{1}{8}}$

Step 8

Simplify.

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Step 8.1

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

$\frac{7}{8} \cdot \frac{1}{x^{\frac{1}{8}}}$

Step 8.2

Multiply $\frac{7}{8}$ by $\frac{1}{x^{\frac{1}{8}}}$ .

$\frac{7}{8 x^{\frac{1}{8}}}$