$100$-th derivative of the function $f(x)=e^{x}\cos(x)$

I’ve got this task I’m not able to solve. So i need to find the 100-th derivative of $$f(x)=e^{x}\cos(x)$$ where $x=\pi$.

I’ve tried using Leibniz’s formula but it got me nowhere, induction doesn’t seem to help either, so if you could just give me a hint, I’d be very grateful.

Many thanks!

Answer

HINT:

$e^x\cos x$ is the real part of $y=e^{(1+i)x}$

As $1+i=\sqrt2e^{i\pi/4}$

$y_n=(1+i)^ne^{(1+i)x}=2^{n/2}e^x\cdot e^{i(n\pi/4+x)}$

Can you take it from here?

Attribution
Source : Link , Question Author : windircurse , Answer Author : lab bhattacharjee

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