Question

Why isn't the chain rule used when differentiating e^-2x for Taylor Series?

Answer 1

what makes you think the chain rule is not used ?

Answer 2

At the beginning (8 minutes in) of lecture 39 (the final review) when the substitute professor writes out the Taylor series for e^x, he lists f'(0)/f''(0)/etc. for e^x for each term as 1, which makes sense. But then when he subs in the term -t^2 for x, he just inserts it directly into the previously derived power series. If the original Taylor series had been written out for -x^2 it seems that each term would have had different coefficients as the result of the change in the various derivatives within the term, since (e^-2x)' gives us -2e^-2x (which evaluated at x=0 should give -2 rather than 1).

Obviously I am missing something, either mixing something up or not accounting for some cancellation of the terms that I am interpreting as missing from the demonstration?

Answer 3

It "turns out" Taylor series has a "substitution property" that lets us do this. For another example, see http://calculus.seas.upenn.edu/?n=Main.ComputingTaylorSeries

However, I have not seen a general proof of this property... but we can show it's true for simple substitutions such as x-> a x^k for constants a and k, by expanding out the Taylor series the "hard way" and showing we get the same series as using substitution.

Answer 4

Thanks, that was very helpful!