Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am studying the generalization/derivation if the $\alpha^{\mbox{th}}$ derivative of $e^{ax}$. I got lost in the third line. Could someone please please fill in the missing lines so that the derivation will be in detail? I really need to figure this thing out. With $\displaystyle D_x^n f(x)=\lim_{h\to 0}h^{-n}\sum_{m=0}^n(-1)^m{}_nC_m f(x+(n-m)h)$ where $_nC_m=\frac{n!}{m!(n-m)!}$ and with $f(x)=e^{ax}$, \begin{array}{rcl} \displaystyle D_x^\alpha e^{ax}&=& \lim_{h\to 0} h^{-\alpha}\sum_{n=0}^\alpha(-1)^n{} _\alpha C_n e^{a(x+(\alpha-n)h)} \\ &=& e^{ax}\lim_{h\to 0} h^{-\alpha}\sum_{n=0}^\alpha(-1)^n{}_\alpha C_n (e^{ah})^{\alpha-n} \\ &=& e^{ax}\lim_{h\to 0} h^{-\alpha} (e^{ah}-1)^\alpha \mbox{what happened???} \\ &=& a^\alpha e^{ax} \end{array}

I didnt get how the third line came up from the second line. Can you plase fill in the missing details for me? Thank you.

share|improve this question
1  
Binomium of Newton? –  Raskolnikov Oct 16 '12 at 8:24
    
@Raskolnikov Yes, but it is probably better known in the English-speaking world as the binomial theorem. –  Harald Hanche-Olsen Oct 16 '12 at 8:39

2 Answers 2

Try this: $$ \sum_{n=0}^{\alpha} \binom{\alpha}{n}(-e^{ah})^{n}=(1-e^{ah})^{\alpha} $$

This is called Binomial theorem. In general: $$ (a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{k}b^{n-k} $$

share|improve this answer

The formula you used is not correct. The correct formula is the Grunwald-Letnikov derivative. I cannot write it in this word processor. Send me a mail and I'll send you a paper with the solution.

mdo@fct.unl.pt

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.