Question: Problem in implementing chain rule of partial differentiation

Initially I have a system of w(x,y,t) and f(x,y,t). I have a set of two partial differential equations with this system. Later on, the system is modified into w(x,y,t,z) andf(x,y,t,z) where x and z are related due to the presence of boundary layers. Because of this, the partial differential expressions with respect to x has to change.

For example, initially: diff(w(x,y,t),x)=diff(w(x,y,t),x)
Later on, when the system gets modified in the presence of boundary layers on application of chain rule: diff(w(x,y,t,z),x)=diff(w(x,y,t,z),x)+diff(w(x,y,t,z),z)*diff(z,x)

This change has to happen for all higher order derivatives of x. In my case, I have upto 4th order derivative in x. How do I achieve these changes in the expressions in Maple in an elegant manner?