Calculus Examples

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

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.6.2

Subtract $3$ from $1$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Factor $3$ out of $- 3$ .

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

Step 2.12

Cancel the common factors.

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Step 2.12.1

Factor $3$ out of $3 x^{\frac{2}{3}}$ .

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

Step 2.12.2

Cancel the common factor.

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

Step 2.12.3

Rewrite the expression.

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

Step 2.13

Move the negative in front of the fraction.

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