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# Find derivatives

If $$f(x) = e^x \sin{x}$$, then $$\frac{\text{d}^{10}}{\text{d} x^{10}} f(x)$$ at $$x=0$$ equals:

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• 10

• 32.

Can you generalise this?

Note by Paramjit Singh
2 years, 11 months ago

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Sure. $$f(x)=e^x\sin x=\Im(e^x(\cos x+i\sin x))=\Im(e^xe^{ix})=\Im(e^{x(1+i)})$$. Hence, $$\frac{d^{n}f}{dx^n}=\Im(\frac{d^n}{dx^n}e^{(1+i)x})=\Im((1+i)^ne^{(1+i)x})$$. We can rewrite $$(1+i)^n=2^{n/2}e^{i\frac{n\pi}{4}}=2^{n/2}(\cos\frac{n\pi}{4}+i\sin\frac{n\pi}{4})$$. When multiplied, we get $$e^{(1+i)x}=1$$ at $$x=0$$, so $$\frac{d^nf}{dx^n}=\Im(2^{n/2}(\cos\frac{n\pi}{4}+i\sin\frac{n\pi}{4}))=\boxed{2^{n/2}\sin\frac{n\pi}{4}}$$. · 2 years, 11 months ago

Lovely! · 2 years, 11 months ago