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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Can you generalise this?

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TopNewestSure. \(f(x)=e^x\sin x=\Im(e^x(\cos x+i\sin x))=\Im(e^xe^{ix})=\Im(e^{x(1+i)})\). 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 \(\frac{d^nf}{dx^n}=\Im(2^{n/2}(\cos\frac{n\pi}{4}+i\sin\frac{n\pi}{4}))=\boxed{2^{n/2}\sin\frac{n\pi}{4}}\).

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Lovely!

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