Precalculus Examples

$f (x) = - 2 \sqrt[4]{x}$

Step 1

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

$\frac{d}{d x} [- 2 x^{\frac{1}{4}}]$

Step 2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{\frac{1}{4}}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{\frac{1}{4}}]$ .

$- 2 \frac{d}{d x} [x^{\frac{1}{4}}]$

Step 3

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

$- 2 (\frac{1}{4} x^{\frac{1}{4} - 1})$

Step 4

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

$- 2 (\frac{1}{4} x^{\frac{1}{4} - 1 \cdot \frac{4}{4}})$

Step 5

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

$- 2 (\frac{1}{4} x^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}})$

Step 6

Combine the numerators over the common denominator.

$- 2 (\frac{1}{4} x^{\frac{1 - 1 \cdot 4}{4}})$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $4$ .

$- 2 (\frac{1}{4} x^{\frac{1 - 4}{4}})$

Step 7.2

Subtract $4$ from $1$ .

$- 2 (\frac{1}{4} x^{\frac{- 3}{4}})$

Step 8

Move the negative in front of the fraction.

$- 2 (\frac{1}{4} x^{- \frac{3}{4}})$

Step 9

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

$- 2 \frac{x^{- \frac{3}{4}}}{4}$

Step 10

Combine $- 2$ and $\frac{x^{- \frac{3}{4}}}{4}$ .

$\frac{- 2 x^{- \frac{3}{4}}}{4}$

Step 11

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

$\frac{- 2}{4 x^{\frac{3}{4}}}$

Step 12

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{4 x^{\frac{3}{4}}}$

Step 13

Cancel the common factors.

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Step 13.1

Factor $2$ out of $4 x^{\frac{3}{4}}$ .

$\frac{2 (- 1)}{2 (2 x^{\frac{3}{4}})}$

Step 13.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 (2 x^{\frac{3}{4}})}$

Step 13.3

Rewrite the expression.

$\frac{- 1}{2 x^{\frac{3}{4}}}$

Step 14

Move the negative in front of the fraction.

$- \frac{1}{2 x^{\frac{3}{4}}}$