I am reading Georgi's book on Lie Algebras in Particle Physics, and there is a formula is his book I cannot derive myself. The formula is for the derivative of a Lie group element with respect to the coordinates as follows:
$$\frac{\partial}{\partial_{\alpha_b}}e^{\alpha_aX_a} = \int_0^1ds\;e^{is\alpha_aX_a}iX_b\;e^{(1-s)\alpha_cX_c}.$$
He claims that this can be derived by writing the exponential as $$e^{i\alpha_aX_a} = \lim_{k\to\infty}\left(1+\frac{\alpha_aX_a}{k}\right)^k,$$ and then taking the derivative of both sides. I cannot make it work, as I get a sum of terms of the form
$$\lim_{k\to\infty}\left(\frac{X_b}{k}\left(1+\frac{\alpha_aX_a}{k}\right)\cdots\left(1+\frac{\alpha_aX_a}{k}\right)+\cdots+\left(1+\frac{\alpha_aX_a}{k}\right)\cdots\left(1+\frac{\alpha_aX_a}{k}\right)\frac{X_b}{k}\right),$$ where the addition denotes all the possible interior derivatives. I cannot see how to rewrite this in the form given. Any help would be much appreciated!
Cheers