Question

Derivative of this following expression...

Answer 1

\[\huge \frac{ d }{ dt } (e ^{-t/\tau} \sin(\omega t))\]

Answer 2

whats so interesting/hard about this

Answer 3

Just getting into the habit of calculus in physics, so I want to make sure I'm doing it right :P... this is my expression \[\huge \frac{ e^{-t/\tau}(\omega t \cos (t \omega) - \sin(t \omega)) }{ \tau }\]

Answer 4

Oh oops that should be a tau not time \[\omega \tau \cos(t \omega)\]

Answer 5

\[\large \begin{align} \left(e ^{-t/\tau} \sin(\omega t)\right)' &=e ^{-t/\tau} \ \left(\sin(\omega t)\right)' + \left(e ^{-t/\tau}\right)' \sin(\omega t) \\~\\ &= e^{-t/\tau } \cos(\omega t)\left( \omega t \right)' + e ^{-t/\tau} \left(-t/\tau \right)' \sin(\omega t) \end{align} \]

Answer 6

Yeah that was it, I couldn't make sense of what I did wrong, and after going over it the 50th time I noticed I had put time there...silly mistake, but thank you ganeshie ^.^

Answer 7

there is a nice pattern when you take nth derivative of \(\large e^xf(x)\)
but with chainrule im not sure if we can avoid the mess

Answer 8

\[\large \dfrac{d^n}{dx^n}e^x f(x) = e^x (1+f)^n\]

Answer 9

or something like that which helps in simplifying the calculation, we can find something nice like above if we spend sometime on chainrule too - e^x is always special !

Answer 10

That's neat, thanks for sharing :P

Answer 11

Check this out http://math.stackexchange.com/questions/18284/nth-derivative-of-e1-x

Answer 12

yeah all of them are variants of leibniz's product rule and binomial theorem

Answer 13

http://en.wikipedia.org/wiki/General_Leibniz_rule

Answer 14

Yeah, it seems really complicated and I've read a bit about Leibniz rule, I'm not sure why but it kind of reminds me of f(x) = lnx finding the nth derivative though obviously it's not as difficult as e^1/x but I still found it neat.

Answer 15

Is there a rule as such for integrating e^x f(x)? I'm assuming by parts would have something to do with it...?

Answer 16

when f(x) = sin/cos, there is a trick

Answer 17

and IBP works when f(x) is polynomial
other than that, there are no rules in general to my knowledge

Answer 18

\[\large \int e^x \sin x ~dx = \mathbb{Img}\int e^{x + i~x} ~dx = \mathbb{Img} \dfrac{e^{x+ix}}{1+i} + C\]

Answer 19

it won't be that easy if f(x) is anything other than sin/cos
you will have to try it by parts before giving up saying the fancy phrase `not possible express in terms of elementary functions` :P