Currently working on a problem sheet that asks us to find the partial derivative $\frac{\partial C}{\partial r}$ of the following formula for the price of a "cash or nothing" call option:
\begin{equation} C(t,S) = Ae^{-rT}\mathbf{N}(d) \end{equation} where,
\begin{align} \mathbf{N}(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^xe^{-\frac{u^2}{2}}\;du && d = \frac{\log(S/E) + (r-\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} \end{align}
I am really stumped and not too sure where to start. I recognise that $\mathbf{N}$ is the normal cumulative distribution function and have tried to start by solving the integral so I can get the terms I need to differentiate out of the integral bounds, however, I have not had any success with this.
Hint: If you view $d$ as a function of $r$ you can compute $d'(r)$ easily and if $f(r) = N(d(r))$ then $f'(r) = N'(d(r)) d'(r)$, so you just need to compute $N'(x)$. Look up the Leibniz rule https://en.wikipedia.org/wiki/Leibniz_integral_rule.