Yes, it's a beautiful formula: log-likelihood function of mvn distribution.
The answer can be found on page 40 of this book: The Matrix Cookbook
You will find equation (81), (57) and (61) are useful to get the partial derivatives.
The partial derivatives are used in Vibrato Monte Carlo method, which is a Path-wise/LRM hybrid method.
Note that there are a few alternative approaches to valuate financial derivatives which have non-differentiable payoff functions.
- Likelihood Ratio Method (LRM)
- Mallianvin Calculus (Stochastic Calculus of Variations)
- "Vibrato" Monte Carlo Method