derivative of composite function and chain rule

Question

I am reading this document:

https://people.math.gatech.edu/~cain/notes/ta.PDF

I made a screenshot of the part of the document that I would like to ask my question on:

I am more or less familiar with the concept of composite function and derivative of such functions using the chain rule. This document mentions it. What I don't understand is how we go from the function $g(t)$ defined as:

$g(t) = f(x(t), y(t), z(t))$

to the derivative of this function defined as:

$g'(t) = \dfrac {\partial f} {\partial x} \bigg\rvert_{P_0}\dfrac {d x} {d t} \bigg\rvert_{t_0} + \dfrac {\partial f} {\partial y} \bigg\rvert_{P_0}\dfrac {d y} {d t} \bigg\rvert_{t_0} + \dfrac {\partial f} {\partial z} \bigg\rvert_{P_0}\dfrac {d z} {d t} \bigg\rvert_{t_0}$

More specifically:

1. why do we use the notation $\dfrac {\partial f} {\partial z}$ and $\dfrac {d x} {d t}$? Why using $\partial$ in one case and $d$ in the other? Why not $\partial$ or $d$ in both cases. Is there a reason for that?
2. why is the derivative a sum of these terms?

Answer 1

This comes from the idea of total derivative versus partial derivative. In words, the total change in a function is equal to the sum of the partial changes. In general, this is related to the Jacobian of the function defined by

$J_{ij}=\frac{\partial f_i}{\partial x_j}$

which satisfies

$f(x+\delta x)-f(x)=J\delta x,$

for $\delta x$ very small. This is a local approximation for the function $f$. The $\partial$ terminology means taking the derivative with all other variables fixed, whereas the $d$ terminology means the total change in the function with respect to a given change in the underlying variables. Because x,y and z are functions of one variable (t), you can use the total derivative, whereas g depends on multiple variables, so you need to have the full equation written down.