# I'm having a problem with the Throat Area formula derivation

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?

• Is that the Travis Taylor book? Oct 14, 2020 at 19:05
• By the same passage, this is apparently "non-trivial" to show. Oct 14, 2020 at 19:13
• @OrganicMarble Yes, that's the one Oct 14, 2020 at 19:16
• What page is this on? I'll need some more context to give a good response. Oct 15, 2020 at 1:23
• I'm guessing it's with respect to gamma (ratio of specific heats, itself a function of P, T, and composition). After differentiating, you'd need to solve for a zero of the differentiated expression, as that would be an extrema (likely, the minimum represented by the throat). You'd then substitute it back in. Oct 15, 2020 at 1:26

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.