It's the chain rule for multivariable functions. Defining a random function $ v(\xi, \eta)$, then $$\frac{\partial v}{\partial x} = \frac{\partial v}{\partial x}\frac{\partial x}{\partial \xi} + \frac{\partial v}{\partial t}\frac{\partial t}{\partial \xi}$$ and using the given x and t functions, you get the resulting partial derivatives of v