find the derivative. g(r) = ln 2r^3 a. g(r) = 1 / 6r^2 b. g(r) = 2r^3 c. g(r) = 1 / 2r^3 d. g(r) = 3 / r

Explanation:

Step1: Use log - property

First, use the property $\ln(ab)=\ln a+\ln b$. So, $g(r)=\ln 2+\ln r^{3}$. Since $\ln 2$ is a constant, its derivative is 0. We focus on $\ln r^{3}$.

Step2: Use power - rule for logarithms

By the power - rule of logarithms $\ln a^{b}=b\ln a$, we have $\ln r^{3} = 3\ln r$.

Step3: Differentiate $\ln r$

The derivative of $\ln x$ with respect to $x$ is $\frac{1}{x}$. So, the derivative of $3\ln r$ with respect to $r$ is $3\times\frac{1}{r}=\frac{3}{r}$.

Answer:

D. $g'(r)=\frac{3}{r}$