Calculus Examples

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

Step 1

Rewrite $\frac{d}{d x} [\frac{1}{\sqrt[3]{x^{4}}}]$ as $\frac{d}{d x} [\frac{1}{x \sqrt[3]{x}}]$ .

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Step 1.1

Factor out $x^{3}$ .

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

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [\frac{1}{x \sqrt[3]{x}}]$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 3

Multiply $x$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 3.1

Multiply $x$ by $x^{\frac{1}{3}}$ .

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Step 3.1.1

Raise $x$ to the power of $1$ .

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

Step 3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2

Write $1$ as a fraction with a common denominator.

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Add $3$ and $1$ .

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

Multiply $4$ by $- 1$ .

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

Step 4.2.3

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $3$ .

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

Step 9.2

Subtract $3$ from $- 4$ .

$- \frac{4}{3} x^{\frac{- 7}{3}}$

Step 10

Move the negative in front of the fraction.

$- \frac{4}{3} x^{- \frac{7}{3}}$

Step 11

Simplify.

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Step 11.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{4}{3} \cdot \frac{1}{x^{\frac{7}{3}}}$

Step 11.2

Combine terms.

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Step 11.2.1

Multiply $\frac{1}{x^{\frac{7}{3}}}$ by $\frac{4}{3}$ .

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

Step 11.2.2

Move $3$ to the left of $x^{\frac{7}{3}}$ .

$- \frac{4}{3 x^{\frac{7}{3}}}$