Calculus Examples

$f (x) = \frac{1}{\sqrt[3]{x^{2}}}$

Step 1

Find the derivative.

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Step 1.1

Apply basic rules of exponents.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}]$

Step 1.1.2

Rewrite $\frac{1}{x^{\frac{2}{3}}}$ as ${(x^{\frac{2}{3}})}^{- 1}$ .

$\frac{d}{d x} [{(x^{\frac{2}{3}})}^{- 1}]$

Step 1.1.3

Multiply the exponents in ${(x^{\frac{2}{3}})}^{- 1}$ .

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Step 1.1.3.1

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

$\frac{d}{d x} [x^{\frac{2}{3} \cdot - 1}]$

Step 1.1.3.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 1.1.3.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$\frac{d}{d x} [x^{\frac{2 \cdot - 1}{3}}]$

Step 1.1.3.2.2

Multiply $2$ by $- 1$ .

$\frac{d}{d x} [x^{\frac{- 2}{3}}]$

Step 1.1.3.3

Move the negative in front of the fraction.

$\frac{d}{d x} [x^{- \frac{2}{3}}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{2}{3}$ .

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

Step 1.3

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

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

Step 1.4

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

$- \frac{2}{3} x^{- \frac{2}{3} + \frac{- 1 \cdot 3}{3}}$

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $3$ .

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

Step 1.6.2

Subtract $3$ from $- 2$ .

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

Step 1.7

Move the negative in front of the fraction.

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

Step 1.8

Simplify.

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Step 1.8.1

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

$- \frac{2}{3} \cdot \frac{1}{x^{\frac{5}{3}}}$

Step 1.8.2

Combine terms.

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Step 1.8.2.1

Multiply $\frac{1}{x^{\frac{5}{3}}}$ by $\frac{2}{3}$ .

$- \frac{2}{x^{\frac{5}{3}} \cdot 3}$

Step 1.8.2.2

Move $3$ to the left of $x^{\frac{5}{3}}$ .

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

Step 2

Set the derivative equal to $0$ then solve the equation $- \frac{2}{3 x^{\frac{5}{3}}} = 0$ .

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Step 2.1

Set the numerator equal to zero.

$2 = 0$

Step 2.2

Since $2 \neq 0$ , there are no solutions.

No solution

Step 3

There are no solution found by setting the derivative equal to $0$ , $- \frac{2}{3 x^{\frac{5}{3}}} = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 4