I came upon this when reading "Introduction to Rocket Science and Engineering, he said "by differentiating" , my question is: differentiate with respect to what? and how do you get such result in the end?
The equation shows that the cross-sectional area of the nozzle depends on the local pressure $P$ and values that are constant. So, he expects you to differentiate with respect to $P$, set the result to zero and solve for the value of $P$. Sounds like a fun project. I would let $x=P/P_c$ and take derivative wrt x.
Some books go through equations with Mach number and finding the throat condition is rather easier. See Mattingly Elements of Gas Turbine Propulsion (1996) in Chapter 3, or Sutton Rocket Propulsion Elements, also Chapter 3.