I am trying to find a way to differentiate the standard normal cumulative distribution function. I think it has something to do with the Leibniz rule but I can't wrap my head around it.

I would like to differentiate N(y) in respect to y, where N(.) denotes the standard normal CDF: $$\frac{d}{dy}N(y) $$ So I think the initial I'd start from the below? $$ \frac{d}{dy}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^ye^{\frac{-x^2}2}dx $$

Any help would be greatly appreciated. Thank you