Calculus Examples

$f (r) = {(2 r)}^{\frac{2}{3}}$ , $r = 5$

Step 1

Find the derivative.

Tap for more steps...

Step 1.1

Simplify with factoring out.

Tap for more steps...

Step 1.1.1

Factor $2$ out of $2 r$ .

$\frac{d}{d r} [{(2 (r))}^{\frac{2}{3}}]$

Step 1.1.2

Apply the product rule to $2 (r)$ .

$\frac{d}{d r} [2^{\frac{2}{3}} r^{\frac{2}{3}}]$

Step 1.2

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

$2^{\frac{2}{3}} \frac{d}{d r} [r^{\frac{2}{3}}]$

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

Tap for more steps...

Step 1.7.1

Multiply $- 1$ by $3$ .

$2^{\frac{2}{3}} (\frac{2}{3} r^{\frac{2 - 3}{3}})$

Step 1.7.2

Subtract $3$ from $2$ .

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

Step 1.8

Move the negative in front of the fraction.

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

Multiply $2^{\frac{2}{3}}$ by $2$ by adding the exponents.

Tap for more steps...

Step 1.11.1

Move $2$ .

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

Step 1.11.2

Multiply $2$ by $2^{\frac{2}{3}}$ .

Tap for more steps...

Step 1.11.2.1

Raise $2$ to the power of $1$ .

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

Step 1.11.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.11.3

Write $1$ as a fraction with a common denominator.

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

Step 1.11.4

Combine the numerators over the common denominator.

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

Step 1.11.5

Add $3$ and $2$ .

$\frac{2^{\frac{5}{3}} r^{- \frac{1}{3}}}{3}$

Step 1.12

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

$\frac{2^{\frac{5}{3}}}{3 r^{\frac{1}{3}}}$

Step 2

Replace the variable $r$ with $5$ in the expression.

$f' (5) = \frac{2^{\frac{5}{3}}}{3 {(5)}^{\frac{1}{3}}}$

Step 3

Remove parentheses.

$f' (5) = \frac{2^{\frac{5}{3}}}{3 \cdot 5^{\frac{1}{3}}}$