I have been assigned to find an 8th order Taylor expansion for an h(x)=f(x)e^g(x), where f(x) is a trig function and g(x) is a power of x. Each derivative expands and after I completed the third derivative I wanted to die. Using the chain and product rule to figure out the 4th may push me over the edge. I am sure there is another way, as my teacher strikes me as far from evil. He also mentioned that the function being even or odd would play a part in some pattern that I may realize... I see that the function is even, but that has not helped me divine any pattern in the subsequent derivatives except that they cycle even/odd.
I wish to become a math wizard, however I am not one currently. For now, I understand basic calculus concepts and operations, so if you can frame your answer with that in mind, it would be immensely appreciated.
I am being vague with the details of the problem here because 1. I don't want to cheat on this assignment (this is for you Prof E., If you are reading this), and 2. I am looking for a general approach to this sort of problem rather than a solution.