What do the partial derivates mean in the formulation of total derivate?

Question

In the total derivate expression of a function $f(x,y,z)$ where $x, y$ and $z$ are functions of $t$, what do the partial derivates $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial z}$ mean? Since $x, y, z$ are functions of $t$, it is impossible to keep one of x, y and z varying and others constant (which is what is done while computing the partially derivatives).

Total derivative of $f$ w.r.t. t: $$ \frac{d f}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial z}\frac{dz}{dt} + \frac{\partial f}{\partial t} $$

Answer 1

You have to consider different cases, don't confuse them :

FIRST CASE : $\quad f(x,y,z)$ $$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz$$

SECOND CASE : $\quad f(x(t),y(t),z(t))$

$dx=\frac{dx}{dt}dt \quad;\quad dy=\frac{dy}{dt}dt \quad;\quad dz=\frac{dz}{dt}dt$ .

$df=\frac{\partial f}{\partial x}\frac{dx}{dt}dt+\frac{\partial f}{\partial y}\frac{dy}{dt}dt+\frac{\partial f}{\partial z}\frac{dz}{dt}dt$

$$\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial z}\frac{dz}{dt}$$

THIRD CASE : $\quad f(x,y,z,t)$ $$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz+\frac{\partial f}{\partial t}dt$$

FOURTH CASE : $\quad f(x(t),y(t),z(t),t)$

$df=\frac{\partial f}{\partial x}\frac{dx}{dt}dt+\frac{\partial f}{\partial y}\frac{dy}{dt}dt+\frac{\partial f}{\partial z}\frac{dz}{dt}dt+\frac{\partial f}{\partial t}dt$

$$\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial z}\frac{dz}{dt}+\frac{\partial f}{\partial t}$$ Don't confuse $\frac{df}{dt}$ and $\frac{\partial f}{\partial t}$

$\frac{\partial f}{\partial t}$ is the partial derivative of $f(x,y,z,t)$ with respect to $t$ with $x,y,z$ not function of $t$.

$\frac{df}{dt}$ is the total derivative of $f(x(t),y(t),z(t),t)$ with $x,y,z$ functions of $t$.

Answer 2

Imagine that initially you only had $f$ depending on $x,y,z$ and you had already computed the partial derivatives $\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}$. Suddenly, you realize that in fact $x,y,z$ depend on a single variable $t$. How to compute $\frac{df}{dt}$ without throwing away the derivatives that you already computed? The answer is given by the formula that you mentioned.