# How do you find the derivative of x^(5/3) * ln(3x)?

May 16, 2016

${x}^{\frac{2}{3}} \left(1 + \frac{5}{3} \ln \left(3 x\right)\right)$

#### Explanation:

differentiate using the $\textcolor{b l u e}{\text{product rule}}$

If f(x) = g(x).h(x) then f'(x) = g(x)h'(x) + h(x)g'(x)...(A)
$\text{-----------------------------------------------------}$

$g \left(x\right) = {x}^{\frac{5}{3}} \Rightarrow g ' \left(x\right) = \frac{5}{3} {x}^{\frac{2}{3}} - \text{using the"color(blue)" power rule}$
$h \left(x\right) = \ln \left(3 x\right) \Rightarrow h ' \left(x\right) = \frac{1}{3 x} \times 3 = \frac{1}{x}$

using $\frac{d}{\mathrm{dx}} \left(\ln \left(f \left(x\right)\right)\right) = \frac{1}{f \left(x\right)} \times \frac{d}{\mathrm{dx}} f \left(x\right)$
$\text{---------------------------------------------------------}$
Substitute these values into (A)

$f ' \left(x\right) = {x}^{\frac{5}{3}} \times \frac{1}{x} + \ln \left(3 x\right) \times \frac{5}{3} {x}^{\frac{2}{3}}$

$= {x}^{\frac{5}{3} - 1} + \frac{5}{3} {x}^{\frac{2}{3}} \ln \left(3 x\right) = {x}^{\frac{2}{3}} + \frac{5}{3} {x}^{\frac{2}{3}} \ln \left(3 x\right)$

Taking out a common factor of ${x}^{\frac{2}{3}}$

$\Rightarrow \frac{d}{\mathrm{dx}} \left({x}^{\frac{5}{3}} . \ln \left(3 x\right)\right) = {x}^{\frac{2}{3}} \left(1 + \frac{5}{3} \ln \left(3 x\right)\right)$