Find derivatives

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

1
-1
10
32.

Can you generalise this?

#Calculus #Derivatives #Goldbach'sConjurersGroup

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 $\frac{d^nf}{dx^n}=\Im(2^{n/2}(\cos\frac{n\pi}{4}+i\sin\frac{n\pi}{4}))=\boxed{2^{n/2}\sin\frac{n\pi}{4}}$ .

Cody Johnson - 7 years, 4 months ago

Lovely!

A Brilliant Member - 7 years, 4 months ago

Carlos Suarez - 7 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$