Calculus Examples

$5 x^{\frac{2}{3}} + 8$

Step 1

By the Sum Rule, the derivative of $5 x^{\frac{2}{3}} + 8$ with respect to $x$ is $\frac{d}{d x} [5 x^{\frac{2}{3}}] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [5 x^{\frac{2}{3}}] + \frac{d}{d x} [8]$

Step 2

Evaluate $\frac{d}{d x} [5 x^{\frac{2}{3}}]$ .

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{2}{3}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$5 \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [8]$

Step 2.2

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

$5 (\frac{2}{3} x^{\frac{2}{3} - 1}) + \frac{d}{d x} [8]$

Step 2.3

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

$5 (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [8]$

Step 2.4

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

$5 (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [8]$

Step 2.5

Combine the numerators over the common denominator.

$5 (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}) + \frac{d}{d x} [8]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

$5 (\frac{2}{3} x^{\frac{2 - 3}{3}}) + \frac{d}{d x} [8]$

Step 2.6.2

Subtract $3$ from $2$ .

$5 (\frac{2}{3} x^{\frac{- 1}{3}}) + \frac{d}{d x} [8]$

Step 2.7

Move the negative in front of the fraction.

$5 (\frac{2}{3} x^{- \frac{1}{3}}) + \frac{d}{d x} [8]$

Step 2.8

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

$5 \frac{2 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [8]$

Step 2.9

Combine $5$ and $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$\frac{5 (2 x^{- \frac{1}{3}})}{3} + \frac{d}{d x} [8]$

Step 2.10

Multiply $2$ by $5$ .

$\frac{10 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [8]$

Step 2.11

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

$\frac{10}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [8]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{10}{3 x^{\frac{1}{3}}} + 0$

Step 3.2

Add $\frac{10}{3 x^{\frac{1}{3}}}$ and $0$ .

$\frac{10}{3 x^{\frac{1}{3}}}$