I don't understand it either- go back and check the problem again!
In particular, "F(x)= ax^2 = ae^TR" makes no sense- it appears to be saying that F(x) is equal to two different functions. The only way I could make sense of it is if they are the same function: x^2= e^(ln(x^2))= e^(2 ln(x)). How is "TR" defined?
In any case "f'(x)= a(n)x^(n-1)" makes no sense because there was no mention of either "f" or "n" before. Assuming you mean "F" and "f" to be the same function (a very bad practice- capital letters and small letters are different symbols and should represent different variables) there still was n "n" in the original formulation.
Of course, it is true that if f(x)= ax^n, not x^2, then f'(x)= a(n x^(n-1)). That is normally proved by using induction on n or the binomial theorem, long before the derivatives of e^x and ln(x) are introduced. But if you do have those, then you could argue that f(x)= ax^n= a e^(n ln(x)) so that f'(x)= a e^(n ln(x))*(n/x)= a(x^n)(n/x)= a(nx^(m-1)).