Calculus Examples

$y = \frac{9}{4 \sqrt[4]{x^{3}}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

$\frac{d}{d x} [\frac{9}{4 x^{\frac{3}{4}}}]$

Step 2

Since $\frac{9}{4}$ is constant with respect to $x$ , the derivative of $\frac{9}{4 x^{\frac{3}{4}}}$ with respect to $x$ is $\frac{9}{4} \frac{d}{d x} [\frac{1}{x^{\frac{3}{4}}}]$ .

$\frac{9}{4} \frac{d}{d x} [\frac{1}{x^{\frac{3}{4}}}]$

Step 3

Apply basic rules of exponents.

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Step 3.1

Rewrite $\frac{1}{x^{\frac{3}{4}}}$ as ${(x^{\frac{3}{4}})}^{- 1}$ .

$\frac{9}{4} \frac{d}{d x} [{(x^{\frac{3}{4}})}^{- 1}]$

Step 3.2

Multiply the exponents in ${(x^{\frac{3}{4}})}^{- 1}$ .

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Step 3.2.1

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

$\frac{9}{4} \frac{d}{d x} [x^{\frac{3}{4} \cdot - 1}]$

Step 3.2.2

Multiply $\frac{3}{4} \cdot - 1$ .

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Step 3.2.2.1

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

$\frac{9}{4} \frac{d}{d x} [x^{\frac{3 \cdot - 1}{4}}]$

Step 3.2.2.2

Multiply $3$ by $- 1$ .

$\frac{9}{4} \frac{d}{d x} [x^{\frac{- 3}{4}}]$

Step 3.2.3

Move the negative in front of the fraction.

$\frac{9}{4} \frac{d}{d x} [x^{- \frac{3}{4}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{3}{4}$ .

$\frac{9}{4} (- \frac{3}{4} x^{- \frac{3}{4} - 1})$

Step 5

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

$\frac{9}{4} (- \frac{3}{4} x^{- \frac{3}{4} - 1 \cdot \frac{4}{4}})$

Step 6

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

$\frac{9}{4} (- \frac{3}{4} x^{- \frac{3}{4} + \frac{- 1 \cdot 4}{4}})$

Step 7

Combine the numerators over the common denominator.

$\frac{9}{4} (- \frac{3}{4} x^{\frac{- 3 - 1 \cdot 4}{4}})$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $4$ .

$\frac{9}{4} (- \frac{3}{4} x^{\frac{- 3 - 4}{4}})$

Step 8.2

Subtract $4$ from $- 3$ .

$\frac{9}{4} (- \frac{3}{4} x^{\frac{- 7}{4}})$

Step 9

Move the negative in front of the fraction.

$\frac{9}{4} (- \frac{3}{4} x^{- \frac{7}{4}})$

Step 10

Combine $x^{- \frac{7}{4}}$ and $\frac{3}{4}$ .

$\frac{9}{4} (- \frac{x^{- \frac{7}{4}} \cdot 3}{4})$

Step 11

Multiply $\frac{9}{4}$ by $\frac{x^{- \frac{7}{4}} \cdot 3}{4}$ .

$- \frac{9 (x^{- \frac{7}{4}} \cdot 3)}{4 \cdot 4}$

Step 12

Multiply.

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Step 12.1

Multiply $3$ by $9$ .

$- \frac{27 x^{- \frac{7}{4}}}{4 \cdot 4}$

Step 12.2

Multiply $4$ by $4$ .

$- \frac{27 x^{- \frac{7}{4}}}{16}$

Step 12.3

Move $x^{- \frac{7}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{27}{16 x^{\frac{7}{4}}}$