TITLE: Proof for the total derivative of a function
QUESTION [11 upvotes]: I'd be interested to understand why is the total derivative of a function $f(t,x,y)$, where $x = x(t)$ and $y=y(t)$ defined as:
$$\frac{df}{dt} = \frac{\partial f}{\partial t}\frac{dt}{dt} + \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$$
This formula seems intuitive to me, but if I would need to prove the general case analytically, how would I do it? :)
Thank you for any help! :)

REPLY [5 votes]: Try starting with f(x+Δx ,y+Δy) - f(x,y) = f(x+Δx, y+∆y) - f(x+Δx, y) + f(x+Δx, y) - f(x, y). 
Reference https://www.physicsforums.com/threads/proof-of-the-total-differential-of-f-x-y.467858/