How to handle expressions with both total derivatives and partial derivatives?

Question

I have a equation with total derivative and partial derivative exist simultaneously like this: this with $y=g(x)$

Why does this equation hold? or More specifically, where this plus comes from?

Answer 1

This follows from the chain rule and the definition of total derivative (which is what $\frac{d}{dx}$ denotes here). https://en.wikipedia.org/wiki/Total_derivative

Technically the $\frac{dg(x)}{dx}$ at the very end of the right hand side is arguably abuse of notation, denoting both total derivative of a function $f$ which is a function of multiple variables, as well as the regular derivative of an implicit function of a single variable, but as one can see on Wikipedia it is standard.

In your case, the Wikipedia $t$ equals your $x$, there is nothing corresponding to the Wikipedia $x$, and the Wikipedia $g(x)$ corresponds to your $y$.

Also the Wikipedia $f$ corresponds to your $\frac{\partial f}{\partial y}$.