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.