Partial Derivative

Written with a curly $\partial$ rather than a straight $d$, as in $\partial f / \partial x$, the notation signals that other variables exist and are being frozen.

The freezing is the entire trick, and it makes the mechanical work no harder than the single-variable case: any term that does not contain the variable you are differentiating is a constant, and constants differentiate to zero. For $g(x, y) = x^2 + 3y^2$, the $3y^2$ term vanishes when differentiating by $x$, leaving $2x$.

This is the unit of measurement for every parameter in a model. "How much would the loss change if I nudged this one weight, leaving all the others alone" is a partial derivative, and collecting one per parameter into a single vector gives you the gradient the optimiser follows.