### Substituting differential equation into an expression

Say I have a (heat) partial differential equation of $f(t,x)$
$$\frac{\partial f}{\partial t}=\frac12\frac{\partial^2 f}{\partial x^2}.$$
I would like to express an algebraic function $g$ of the variable $\Big(\frac{\partial f}{\partial t}, \frac{\partial^2 f}{\partial t\partial x}\Big)$ all in the partial derivatives of $x$. That is
$$\frac{\partial^2 f}{\partial t\partial x}=\frac12\frac{\partial^3 f}{\partial x^3}.$$
The result should be
$$g\Big(\frac{\partial f}{\partial t}, \frac{\partial^2 f}{\partial t\partial x}\Big)=g\Big(\frac12\frac{\partial^2 f}{\partial x^2}, \frac12\frac{\partial^3 f}{\partial x^3}\Big).$$

How can I accomplish this with SageMath?