Derivative for log

I have the following problem:

$\mathrm{log}\left(\frac{x+3}{4-x}\right)$

I need to graph the following function so I will need a starting point, roots, zeros, stationary points, inflection points and local minimum and maximum and I need to know where the function grows and declines.

I calculated roots zeros $x+3=0,x=-3$ and roots $4-x=0,x=-4$. Now I sort of know how to graph the function from here but how do I get the stationary points do I have to find the derivative of $\mathrm{log}\left(\frac{x+3}{4-x}\right)$ or just $\left(\frac{x+3}{4-x}\right)$

I don't fully understand how to find the derivative of log. Can i use the $\mathrm{log}\left(x\right)}^{\prime}=\frac{1}{x$ rule here to get $\frac{1}{\frac{x+3}{4-x}}$ and then find stationary points here ?