Understanding a notation chain rule for multivariable functions

Question

I can't understand the meaning of partial derivative times differential. I was reading wikipedia and poped to Total derivative article, Where I saw this:

The total derivative of $ {\displaystyle f(t,x(t),y(t))}$ with respect to ${\displaystyle t}$ is

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

which can be simplefied to:

${\displaystyle \operatorname {d} f={\frac {\partial f}{\partial t}}\operatorname {d} t+{\frac {\partial f}{\partial x}}\operatorname {d} x+{\frac {\partial f}{\partial y}}\operatorname {d} y}$

What does it mean to take a partial derivative ${\frac {\partial f}{\partial t}}$ (which is a new function by itself) and multiply it by a differential $\operatorname {d}t$.

I'm in high school currently and I read what interests me. I don't have an comprehensive knowledge, therefore it might have been taught in topics which I have'nt learned. So any question will be welcomed :)

Answer 1

Let us say, just for some toy example, that $z(t) = t$, $x(t) = t^2$ and $y(t) =2t$ where $f(z, x, y) = xyz$, thus $f( t, x(t), y(t) ) = t\times t^2\times 2t = 2t^4.$ Now, you want find $f_t'$, clearly $(2t^4)' = 8t^3$, or alternatively you can use the chain rule \begin{align} f'_t &=f'_tz_t' + f'_xx_t' + f'_yy_t'\\ &= x(t)y(t)t'+y(t)z(t)x'(t)+x(t)z(t)y'(t)\\ &= 2t^3+4t^3+2t^3\\ &=8t^3. \end{align}

Answer 2

Maybe a coordinate-free explanation might be helpful. We have a path $c$ defined on an interval $J\to\mathbb R^n$ and a real valued function $f\colon\mathbb R^n\to\mathbb R$. Then the derivative of $f\circ c\colon J\to\mathbb R$ at $t\in J$ may be calculated as usual via chain rule: multiply the derivative of $f$ at $c(t)$ with the derivative of $c$ at $t$, that is

$$f’(t)=\langle \nabla f\bigl(c(t)\bigr),c’(t)\rangle$$

or in short $$f’=\langle(\nabla f)\circ c,c’\rangle.$$