Will an equation still be valid if we take total derivative of both side? What about partial derivative instead?
For example, suppose we have $\varphi (s, \varphi(t,x)) = \varphi(s+t,x)$, does that mean $\frac{\partial \varphi}{\partial t} (s, \varphi(t,x)) = \frac{\partial \varphi}{\partial t} (s+t,x)$? What about $\frac{d \varphi}{ds} (s, \varphi(t,x)) = \frac{d \varphi}{d s} (s+t,x)$?