Calculus Examples

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

Step 1

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

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{\frac{1}{3}}}$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Move the negative in front of the fraction.

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

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

Step 7.2

Subtract $3$ from $- 1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

Simplify the expression.

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Step 8.3.1

Multiply $- 1$ by $- 1$ .

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

Step 8.3.2

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

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

Step 8.3.3

Move $x^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{1}{3 x^{\frac{4}{3}}} + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 10

Simplify the expression.

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Step 10.1

Multiply $\frac{1}{x^{\frac{1}{3}}}$ by $0$ .

$\frac{1}{3 x^{\frac{4}{3}}} + 0$

Step 10.2

Add $\frac{1}{3 x^{\frac{4}{3}}}$ and $0$ .

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