Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\sqrt[3]{x} - \frac{1}{\sqrt[3]{x}}$ with respect to $x$ is $\frac{d}{d x} [\sqrt[3]{x}] + \frac{d}{d x} [- \frac{1}{\sqrt[3]{x}}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\sqrt[3]{x}]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.6.2

Subtract $3$ from $1$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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 $g (x) = \frac{1}{x^{\frac{1}{3}}}$ .

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{3}}$ .

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

Step 3.4.2

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

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

Step 3.4.3

Replace all occurrences of $u$ with $x^{\frac{1}{3}}$ .

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

Step 3.5

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

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

Step 3.6

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

$\frac{1}{3} x^{- \frac{2}{3}} - (- {(x^{\frac{1}{3}})}^{- 2} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.7

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

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Step 3.7.1

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

$\frac{1}{3} x^{- \frac{2}{3}} - (- x^{\frac{1}{3} \cdot - 2} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.7.2

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

$\frac{1}{3} x^{- \frac{2}{3}} - (- x^{\frac{- 2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.7.3

Move the negative in front of the fraction.

$\frac{1}{3} x^{- \frac{2}{3}} - (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.8

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

$\frac{1}{3} x^{- \frac{2}{3}} - (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.9

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

$\frac{1}{3} x^{- \frac{2}{3}} - (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.10

Combine the numerators over the common denominator.

$\frac{1}{3} x^{- \frac{2}{3}} - (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.11

Simplify the numerator.

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Step 3.11.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} x^{- \frac{2}{3}} - (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 3}{3}})) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.11.2

Subtract $3$ from $1$ .

$\frac{1}{3} x^{- \frac{2}{3}} - (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{- 2}{3}})) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.12

Move the negative in front of the fraction.

$\frac{1}{3} x^{- \frac{2}{3}} - (- x^{- \frac{2}{3}} (\frac{1}{3} x^{- \frac{2}{3}})) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.13

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

$\frac{1}{3} x^{- \frac{2}{3}} - (- x^{- \frac{2}{3}} \frac{x^{- \frac{2}{3}}}{3}) + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.14

Combine $\frac{x^{- \frac{2}{3}}}{3}$ and $x^{- \frac{2}{3}}$ .

$\frac{1}{3} x^{- \frac{2}{3}} - - \frac{x^{- \frac{2}{3}} x^{- \frac{2}{3}}}{3} + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.15

Multiply $x^{- \frac{2}{3}}$ by $x^{- \frac{2}{3}}$ by adding the exponents.

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Step 3.15.1

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

$\frac{1}{3} x^{- \frac{2}{3}} - - \frac{x^{- \frac{2}{3} - \frac{2}{3}}}{3} + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.15.2

Combine the numerators over the common denominator.

$\frac{1}{3} x^{- \frac{2}{3}} - - \frac{x^{\frac{- 2 - 2}{3}}}{3} + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.15.3

Subtract $2$ from $- 2$ .

$\frac{1}{3} x^{- \frac{2}{3}} - - \frac{x^{\frac{- 4}{3}}}{3} + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.15.4

Move the negative in front of the fraction.

$\frac{1}{3} x^{- \frac{2}{3}} - - \frac{x^{- \frac{4}{3}}}{3} + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.16

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

$\frac{1}{3} x^{- \frac{2}{3}} - - \frac{1}{3 x^{\frac{4}{3}}} + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.17

Multiply $- 1$ by $- 1$ .

$\frac{1}{3} x^{- \frac{2}{3}} + 1 \frac{1}{3 x^{\frac{4}{3}}} + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.18

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

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{1}{3 x^{\frac{4}{3}}} + \frac{1}{x^{\frac{1}{3}}} \cdot 0$

Step 3.19

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

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{1}{3 x^{\frac{4}{3}}} + 0$

Step 3.20

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

$\frac{1}{3} x^{- \frac{2}{3}} + \frac{1}{3 x^{\frac{4}{3}}}$

Step 4

Simplify.

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Step 4.1

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

$\frac{1}{3} \cdot \frac{1}{x^{\frac{2}{3}}} + \frac{1}{3 x^{\frac{4}{3}}}$

Step 4.2

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

$\frac{1}{3 x^{\frac{2}{3}}} + \frac{1}{3 x^{\frac{4}{3}}}$