Partial derivative operator expansion

Question

Expand $$ (h\frac{\partial}{\partial t}+hf\frac{\partial}{\partial x})^3 $$ Where $$ x'(t)=f(x,t)=f $$ If this question is ridiculously hard to answer, at least tell me if this is correct $$ (h\frac{\partial}{\partial t}+hf\frac{\partial}{\partial x})^2=h^2(\frac{\partial^2}{\partial x^2}+\frac{\partial f}{\partial t}\frac{\partial}{\partial x}+f\frac{\partial^2}{\partial t \partial x}+f\frac{\partial^2}{\partial x \partial t}+f^2\frac{\partial^2}{\partial x^2}) $$ And is this also correct with function g $$ f\frac{\partial^2}{\partial t \partial x}g \neq f\frac{\partial^2}{\partial x \partial t}g $$

Answer 1

$$ (h\frac{\partial}{\partial t}+hf\frac{\partial}{\partial x})^3\\ \implies h^3\frac{\partial^3}{\partial t^3}+3h^2\frac{\partial^2}{\partial t^2}hf\frac{\partial}{\partial x}+3h\frac{\partial}{\partial t}h^2f^2\frac{\partial^2}{\partial x^2}+h^3f^3\frac{\partial^3}{\partial x^3} $$ I'll leave it to you to use the chain rule, and the given information.