Calculus Examples

$y = \sqrt[4]{5 x}$

Step 1

Simplify with factoring out.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5 x}$ as ${(5 x)}^{\frac{1}{4}}$ .

$\frac{d}{d x} [{(5 x)}^{\frac{1}{4}}]$

Step 1.2

Factor $5$ out of $5 x$ .

$\frac{d}{d x} [{(5 (x))}^{\frac{1}{4}}]$

Step 1.3

Apply the product rule to $5 (x)$ .

$\frac{d}{d x} [5^{\frac{1}{4}} x^{\frac{1}{4}}]$

Step 2

Since $5^{\frac{1}{4}}$ is constant with respect to $x$ , the derivative of $5^{\frac{1}{4}} x^{\frac{1}{4}}$ with respect to $x$ is $5^{\frac{1}{4}} \frac{d}{d x} [x^{\frac{1}{4}}]$ .

$5^{\frac{1}{4}} \frac{d}{d x} [x^{\frac{1}{4}}]$

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $4$ .

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

Step 7.2

Subtract $4$ from $1$ .

$5^{\frac{1}{4}} (\frac{1}{4} x^{\frac{- 3}{4}})$

Step 8

Move the negative in front of the fraction.

$5^{\frac{1}{4}} (\frac{1}{4} x^{- \frac{3}{4}})$

Step 9

Combine $\frac{1}{4}$ and $x^{- \frac{3}{4}}$ .

$5^{\frac{1}{4}} \frac{x^{- \frac{3}{4}}}{4}$

Step 10

Combine $5^{\frac{1}{4}}$ and $\frac{x^{- \frac{3}{4}}}{4}$ .

$\frac{5^{\frac{1}{4}} x^{- \frac{3}{4}}}{4}$

Step 11

Move $x^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{5^{\frac{1}{4}}}{4 x^{\frac{3}{4}}}$