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Real Analysis : Differentiability and Sequence of Partial Sums

A). Prove that the function f(x) = e^x is differentiable on R, and that (e^x)' = e^x. ( Hint: Use the definition of e^x, and consider the sequence of partial sums.)

My thoughts on a:
I tried to prove the differentiability by proving continuity on R, since e^x is series, sum of polynomials, and all polynomials are differentiable on R, so is their sum. But how to prove that, since the sum goes to infinity, and we cannot do induction on n going to infinity? Also I am trying to use the hint, but how would the sequence of partial sums apply here? I need detailed answer and proof of every claim made, all by using the def of e^x only.

b). Prove that the function f(x) = cos x is differentiable on R, and that (cos x)' = - sin x, and prove that f (x) = sin x is differentiable on R, and that (sin x)' = cos x, of course here part b is also by using def of sin and cos as functions of series..please justify every step clearly. Thanks in advance.

Since the procedure of a and b is very similar, and since one can tell that I understand the problem and how to solve it, but have some doubts and want to double check, I believe 15 mins will be enough to solve a and b. I give one credit.

Solution Preview

I am presuming that the definitions you refer to are the usual Taylor series expansions.
Because the people on the net don't have your text and notes, it helps if you specify things like this, thanks.

Yes, you are absolutely on the right track here.

In fact, the Taylor series are developed by using derivatives, so this question is highly circular logic and doesn't really class as a proof. However, we're asked to do the demonstration so we do it.

Yes, we can do proofs on a sum going to ...

Solution Summary

Differentiability and Sequence of Partial Sums are investigated.