I'm studying a text on the spherical Bessel functions and I've come across the following integral: $$(-z)^l\left(\frac{1}{z}\frac{d}{dz}\right)^l\frac{1}{2}\int_{-1}^{1} e^{izx} \,dx = \frac{z^l}{2}\int_{-1}^{1} \frac{(1-x^2)^l}{2^ll!}e^{izx} \,dx$$
I've been trying to derive the result on the right hand side by repeatedly taking the derivative with respect to $z$ under the integral but haven't been successful. Am I correct in writing that $$\frac{d}{dz}\int_{-1}^{1} e^{izx} \,dx = \int_{-1}^{1} ixe^{izx} \,dx$$ If not would someone be able to show me how to take the derivative correctly? Thanks.