Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Cancel the common factor.

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

Step 12

Rewrite the expression.

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