Calculus Examples

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

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $3$ .

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

Step 2.7.2

Subtract $3$ from $1$ .

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

Step 2.8

Move the negative in front of the fraction.

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

Step 2.9

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

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

Step 2.10

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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