Suppose:
f = a + b g[x, y]
h = D[f, {x, 2}, {y, 1}]
(*Out:= b g^(2,1)[x,y]*)
In the function $h=b\frac{\partial^3g}{\partial x^2\partial y}$ I want to replace $\frac{\partial g}{\partial y}$ by $c\frac{\partial u}{\partial t}$ to obtain $h=bc\frac{\partial^3u}{\partial x^2\partial t}$.
In the function $h=b\frac{\partial^3g}{\partial x^2\partial y}$ I want to replace $\frac{\partial^2 g}{\partial x^2}$ by $c\frac{\partial u}{\partial t}$ to obtain $h=bc\frac{\partial^2u}{\partial y\partial t}$.
After looking at the Fullform
of partial derivatives this seems like a hopeless task. I have also read this post but it hasn't helped. Any suggestions and help is much appreciated.