Question

find the derivative of the function $f(x)=log_3left(\frac{3x^{2}}{sqrt{x}}
ight)$. select the correct answer below: $\frac{1}{2xln(3)}$, $\frac{3}{2xln(3)}$, $-\frac{1}{2xln(3)}$, $-\frac{2}{3xln(3)}$

Explanation:

Step1: Simplify the function

$f(x)=\log_3(3x^{2-\frac{1}{2}})=\log_3(3x^{\frac{3}{2}})=1 + \frac{3}{2}\log_3(x)$

Step2: Differentiate term - by - term

The derivative of a constant is 0, and the derivative of $\log_a(x)$ is $\frac{1}{x\ln(a)}$. So $f^\prime(x)=0+\frac{3}{2}\times\frac{1}{x\ln(3)}=\frac{3}{2x\ln(3)}$

Answer:

$\frac{3}{2x\ln(3)}$ (corresponding to the second option in the multiple - choice)