Calculus Examples

$f (x) = \sqrt[4]{x^{5}}$

Step 1

Rewrite $\frac{d}{d x} [\sqrt[4]{x^{5}}]$ as $\frac{d}{d x} [x \sqrt[4]{x}]$ .

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Step 1.1

Factor out $x^{4}$ .

$\frac{d}{d x} [\sqrt[4]{x^{4} x}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [x \sqrt[4]{x}]$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$\frac{d}{d x} [x \cdot x^{\frac{1}{4}}]$

Step 3

Multiply $x$ by $x^{\frac{1}{4}}$ by adding the exponents.

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Step 3.1

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

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Step 3.1.1

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

$\frac{d}{d x} [x^{1} x^{\frac{1}{4}}]$

Step 3.1.2

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

$\frac{d}{d x} [x^{1 + \frac{1}{4}}]$

Step 3.2

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

$\frac{d}{d x} [x^{\frac{4}{4} + \frac{1}{4}}]$

Step 3.3

Combine the numerators over the common denominator.

$\frac{d}{d x} [x^{\frac{4 + 1}{4}}]$

Step 3.4

Add $4$ and $1$ .

$\frac{d}{d x} [x^{\frac{5}{4}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{4}$ .

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

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 6

Combine $- 1$ and $\frac{4}{4}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $4$ .

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

Step 8.2

Subtract $4$ from $5$ .

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

Step 9

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

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