Calculus Examples

$\frac{4}{3 \sqrt[5]{x^{3}}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{3}}$ as $x^{\frac{3}{5}}$ .

$\frac{d}{d x} [\frac{4}{3 x^{\frac{3}{5}}}]$

Step 2

Since $\frac{4}{3}$ is constant with respect to $x$ , the derivative of $\frac{4}{3 x^{\frac{3}{5}}}$ with respect to $x$ is $\frac{4}{3} \frac{d}{d x} [\frac{1}{x^{\frac{3}{5}}}]$ .

$\frac{4}{3} \frac{d}{d x} [\frac{1}{x^{\frac{3}{5}}}]$

Step 3

Apply basic rules of exponents.

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Step 3.1

Rewrite $\frac{1}{x^{\frac{3}{5}}}$ as ${(x^{\frac{3}{5}})}^{- 1}$ .

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

Step 3.2

Multiply the exponents in ${(x^{\frac{3}{5}})}^{- 1}$ .

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Step 3.2.1

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

$\frac{4}{3} \frac{d}{d x} [x^{\frac{3}{5} \cdot - 1}]$

Step 3.2.2

Multiply $\frac{3}{5} \cdot - 1$ .

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Step 3.2.2.1

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

$\frac{4}{3} \frac{d}{d x} [x^{\frac{3 \cdot - 1}{5}}]$

Step 3.2.2.2

Multiply $3$ by $- 1$ .

$\frac{4}{3} \frac{d}{d x} [x^{\frac{- 3}{5}}]$

Step 3.2.3

Move the negative in front of the fraction.

$\frac{4}{3} \frac{d}{d x} [x^{- \frac{3}{5}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{3}{5}$ .

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

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 6

Combine $- 1$ and $\frac{5}{5}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $5$ .

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

Step 8.2

Subtract $5$ from $- 3$ .

$\frac{4}{3} (- \frac{3}{5} x^{\frac{- 8}{5}})$

Step 9

Move the negative in front of the fraction.

$\frac{4}{3} (- \frac{3}{5} x^{- \frac{8}{5}})$

Step 10

Combine $x^{- \frac{8}{5}}$ and $\frac{3}{5}$ .

$\frac{4}{3} (- \frac{x^{- \frac{8}{5}} \cdot 3}{5})$

Step 11

Multiply $\frac{4}{3}$ by $\frac{x^{- \frac{8}{5}} \cdot 3}{5}$ .

$- \frac{4 (x^{- \frac{8}{5}} \cdot 3)}{3 \cdot 5}$

Step 12

Multiply.

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Step 12.1

Multiply $3$ by $4$ .

$- \frac{12 x^{- \frac{8}{5}}}{3 \cdot 5}$

Step 12.2

Multiply $3$ by $5$ .

$- \frac{12 x^{- \frac{8}{5}}}{15}$

Step 12.3

Move $x^{- \frac{8}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{12}{15 x^{\frac{8}{5}}}$

Step 13

Factor $3$ out of $12$ .

$- \frac{3 (4)}{15 x^{\frac{8}{5}}}$

Step 14

Cancel the common factors.

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Step 14.1

Factor $3$ out of $15 x^{\frac{8}{5}}$ .

$- \frac{3 (4)}{3 (5 x^{\frac{8}{5}})}$

Step 14.2

Cancel the common factor.

$- \frac{3 \cdot 4}{3 (5 x^{\frac{8}{5}})}$

Step 14.3

Rewrite the expression.

$- \frac{4}{5 x^{\frac{8}{5}}}$