Partial derivative of 3D function with time

Question

Having a little bit of trouble wrapping my head round this one. I have an undefined function of $\phi (x,y,z,t)$ and its partial derivatives $\phi_x$ and $\phi_t$, and i want to find $\frac{\partial}{\partial x} \phi_x \phi_t$. My thoughts were to use chain rule to obtain $\phi_{xx} \phi_t + \phi_x \phi_{tx}$, but am not sure if this is correct. Any help would be greatly appreciated.

Answer 1

When taking partial derivatives, you consider the "other" variables as constants. Hence,

$$\phi_x(x,y,z,t)=f(x)$$ and $$\phi_t(x,y,z,t)=g(x).$$

Then by the chain rule

$$(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)$$ where

$$f'(x)=\phi_{xx}(x,y,z,t)$$ and $$g'(x)=\phi_{tx}(x,y,z,t).$$