Calculus Examples

$f (x) = 2 x + 3 x^{\frac{2}{3}}$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $3$ .

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

Step 3.6.2

Subtract $3$ from $2$ .

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

Step 3.7

Move the negative in front of the fraction.

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Multiply $2$ by $3$ .

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

Step 3.11

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

$2 + \frac{6}{3 x^{\frac{1}{3}}}$

Step 3.12

Factor $3$ out of $6$ .

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

Step 3.13

Cancel the common factors.

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Step 3.13.1

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

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

Step 3.13.2

Cancel the common factor.

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

Step 3.13.3

Rewrite the expression.

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