Precalculus Examples

$f (x) = \sqrt[5]{x^{6}}$

Step 1

Rewrite $\frac{d}{d x} [\sqrt[5]{x^{6}}]$ as $\frac{d}{d x} [x \sqrt[5]{x}]$ .

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Step 1.1

Factor out $x^{5}$ .

$\frac{d}{d x} [\sqrt[5]{x^{5} x}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [x \sqrt[5]{x}]$

Step 2

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

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

Step 3

Multiply $x$ by $x^{\frac{1}{5}}$ by adding the exponents.

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Step 3.1

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

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Step 3.1.1

Raise $x$ to the power of $1$ .

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

Step 3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2

Write $1$ as a fraction with a common denominator.

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Add $5$ and $1$ .

$\frac{d}{d x} [x^{\frac{6}{5}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{6}{5}$ .

$\frac{6}{5} x^{\frac{6}{5} - 1}$

Step 5

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

$\frac{6}{5} x^{\frac{6}{5} - 1 \cdot \frac{5}{5}}$

Step 6

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

$\frac{6}{5} x^{\frac{6}{5} + \frac{- 1 \cdot 5}{5}}$

Step 7

Combine the numerators over the common denominator.

$\frac{6}{5} x^{\frac{6 - 1 \cdot 5}{5}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $5$ .

$\frac{6}{5} x^{\frac{6 - 5}{5}}$

Step 8.2

Subtract $5$ from $6$ .

$\frac{6}{5} x^{\frac{1}{5}}$

Step 9

Combine $\frac{6}{5}$ and $x^{\frac{1}{5}}$ .

$\frac{6 x^{\frac{1}{5}}}{5}$