Expansion of Second Derivative (multiple variables)

Question

I'm looking for a way to express $\dfrac{\partial^2 f}{\partial \mu \,\partial \nu}$ in terms of $\dfrac{\partial^2 f}{\partial x_i \,\partial x_j}$ and $\dfrac{\partial f}{\partial x_i}$. My attempt is shown below, is this derivation correct? $$ \begin{align} \frac{\partial^2 f}{\partial \mu \,\partial \nu} & = \frac{\partial}{\partial \mu} \sum^{}_{i} \frac{\partial f}{\partial x_i} \frac{\partial x_i}{\partial \nu}\\ & = \sum^{}_{i} \left( \frac{\partial x_i}{\partial \nu} \frac{\partial^2 f}{\partial \mu \,\partial x_i} +\frac{\partial f}{\partial x_i} \frac{\partial^2 x_i}{\partial \mu \,\partial \nu} \right)\\ & = \sum^{}_{i} \left(\frac{\partial x_i}{\partial \nu} \frac{\partial}{\partial x_i}\sum^{}_{j} \frac{\partial f}{\partial x_j} \frac{\partial x_j}{\partial \mu} + \frac{\partial f}{\partial x_i} \frac{\partial^2 x_i}{\partial \mu \,\partial \nu} \right)\\ & = \sum^{}_{i} \left(\frac{\partial x_i}{\partial \nu} \sum^{}_{j} \left( \frac{\partial x_j}{\partial \mu} \frac{\partial^2 f}{\partial x_i \,\partial x_j} +\frac{\partial f}{\partial x_j} \frac{\partial^2 x_j}{\partial x_i \,\partial \mu} \right) + \frac{\partial f}{\partial x_i} \frac{\partial^2 x_i}{\partial \mu\, \partial \nu} \right)\\ \end{align} $$