Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $3 x^{\frac{2}{3}} - 2 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{\frac{2}{3}}] + \frac{d}{d x} [- 2 x]$ .

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

Step 2

Evaluate $\frac{d}{d x} [3 x^{\frac{2}{3}}]$ .

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{\frac{2}{3}}$ with respect to $x$ is $3 \frac{d}{d x} [x^{\frac{2}{3}}]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.6.2

Subtract $3$ from $2$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Multiply $2$ by $3$ .

$\frac{6 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- 2 x]$

Step 2.11

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

$\frac{6}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- 2 x]$

Step 2.12

Factor $3$ out of $6$ .

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

Step 2.13

Cancel the common factors.

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Step 2.13.1

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

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

Step 2.13.2

Cancel the common factor.

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

Step 2.13.3

Rewrite the expression.

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

Step 3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

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

Step 3.2

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

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

Step 3.3

Multiply $- 2$ by $1$ .

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