Calculus Examples

$y = \frac{- 6}{\sqrt[4]{x}}$

Step 1

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

$y = \frac{- 6}{x^{\frac{1}{4}}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{- 6}{x^{\frac{1}{4}}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{6}{x^{\frac{1}{4}}}]$

Step 4.2

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6}{x^{\frac{1}{4}}}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{1}{x^{\frac{1}{4}}}]$ .

$- 6 \frac{d}{d x} [\frac{1}{x^{\frac{1}{4}}}]$

Step 4.3

Apply basic rules of exponents.

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Step 4.3.1

Rewrite $\frac{1}{x^{\frac{1}{4}}}$ as ${(x^{\frac{1}{4}})}^{- 1}$ .

$- 6 \frac{d}{d x} [{(x^{\frac{1}{4}})}^{- 1}]$

Step 4.3.2

Multiply the exponents in ${(x^{\frac{1}{4}})}^{- 1}$ .

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Step 4.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 6 \frac{d}{d x} [x^{\frac{1}{4} \cdot - 1}]$

Step 4.3.2.2

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

$- 6 \frac{d}{d x} [x^{\frac{- 1}{4}}]$

Step 4.3.2.3

Move the negative in front of the fraction.

$- 6 \frac{d}{d x} [x^{- \frac{1}{4}}]$

Step 4.4

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

$- 6 (- \frac{1}{4} x^{- \frac{1}{4} - 1})$

Step 4.5

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

$- 6 (- \frac{1}{4} x^{- \frac{1}{4} - 1 \cdot \frac{4}{4}})$

Step 4.6

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

$- 6 (- \frac{1}{4} x^{- \frac{1}{4} + \frac{- 1 \cdot 4}{4}})$

Step 4.7

Combine the numerators over the common denominator.

$- 6 (- \frac{1}{4} x^{\frac{- 1 - 1 \cdot 4}{4}})$

Step 4.8

Simplify the numerator.

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Step 4.8.1

Multiply $- 1$ by $4$ .

$- 6 (- \frac{1}{4} x^{\frac{- 1 - 4}{4}})$

Step 4.8.2

Subtract $4$ from $- 1$ .

$- 6 (- \frac{1}{4} x^{\frac{- 5}{4}})$

Step 4.9

Move the negative in front of the fraction.

$- 6 (- \frac{1}{4} x^{- \frac{5}{4}})$

Step 4.10

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

$- 6 (- \frac{x^{- \frac{5}{4}}}{4})$

Step 4.11

Multiply $- 1$ by $- 6$ .

$6 \frac{x^{- \frac{5}{4}}}{4}$

Step 4.12

Combine $6$ and $\frac{x^{- \frac{5}{4}}}{4}$ .

$\frac{6 x^{- \frac{5}{4}}}{4}$

Step 4.13

Move $x^{- \frac{5}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{6}{4 x^{\frac{5}{4}}}$

Step 4.14

Factor $2$ out of $6$ .

$\frac{2 (3)}{4 x^{\frac{5}{4}}}$

Step 4.15

Cancel the common factors.

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Step 4.15.1

Factor $2$ out of $4 x^{\frac{5}{4}}$ .

$\frac{2 (3)}{2 (2 x^{\frac{5}{4}})}$

Step 4.15.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 (2 x^{\frac{5}{4}})}$

Step 4.15.3

Rewrite the expression.

$\frac{3}{2 x^{\frac{5}{4}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{3}{2 x^{\frac{5}{4}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{3}{2 x^{\frac{5}{4}}}$