Find n-th derivative using taylor expansion

Asked Apr 09 '19 at 11:56

Active Apr 09 '19 at 12:13

Viewed 55 times

Find the n-th derivative of the following functions:

a) $f(x)= e^{x^2}$ b) $f(x)= e^{\frac{a}{x}}$ c) $f(x)= \arctan{x}$

Using

$f(x+h)-f(x) = h f{'}(x) + \frac{h^2}{2!}f{''}(x) + ...$

My initial idea was to find some recursive pattern but didn't have any luck with that.

edited Apr 09 '19 at 12:13

Eleven-Eleven

asked Apr 09 '19 at 11:56

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Is $e^{a/c}$ intended ? – Apr 09 '19 at 11:58
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Don't you mean derivative at a certain point ? – Apr 09 '19 at 11:59
sorry it is $e^{\frac{a}{x}}$ – Apr 09 '19 at 11:59
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You might like to have a read of these pages: Find an expression for the $n$-th derivative of $f(x)=e^{x^2}$, Closed form of the $n^{th}$-derivative of $f(x) = \arctan x$, $n$th derivative of $e^{1/x}$. – Minus One-Twelfth Apr 09 '19 at 12:13
The last link above discusses the $n$-th derivative of $e^{1/x}$ rather than $e^{a/x}$, but you can use it with the chain rule to get what you want. That is, using the fact that $\frac{d^n}{dx^n}(f(bx)) = b^n f^{(n)}(bx)$ (via repeated chain rule), if you know an expression for the $n$-th derivative of $f(x):=e^{1/x}$, say $f^{(n)}(x) = g_n(x)$, then you can use this to get an expression for the $n$-th derivative of $e^{a/x} = e^{1/(x/a)}=f\left(a^{-1}x\right)$, namely $\frac{d^n}{dx^n}\left(e^{a/x}\right) = a^{-n} g_n\left(a^{-1}x\right)$. – Minus One-Twelfth Apr 09 '19 at 12:23
Awesome thank you so much! – Apr 09 '19 at 12:27

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