Differentiating the Risk Neutral Price of a Call Option

I know that the risk-neutral price of a call option with strike price $K$ is given by: $C(K,S_T) = e^{-rT}\int_{0}^{\infty }(S_T-K)^+g(S_T)dS_t$ Since a payoff is only valid when $S_T >K$ this turns into: $C(K,S_T) = e^{-rT}\int_{K}^{\infty }(S_T-K)\times g(S_T)dS_t$ Now I wanted to differentiate the call formula once and twice with respect to K, so $\frac{\partial C}{\partial K}$ and $\frac{\partial ^2C}{\partial K^2}$ but this is where I am a little stuck. Would I use Leibniz integra