How do we obtain $df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$ from $f(x,y)$?

Question

See this: What is the Jacobian, how does it work, and what is an intuitive explanation of the Jacobian and a change of basis?

Consider a function $f(x,y)$. The differential $df$ is given by: $df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$
This looks a lot like the gradient equation: $\nabla f = \frac{\partial f}{\partial x} {i} + \frac{\partial f}{\partial y}{ j}$

This is the basic idea behind the Jacobian (and differential forms) - that a differential can be viewed as a sort of vector, with its components being given by the partial derivative terms and the ‘basis’ as the differentials of the independent variables - $dx$ and $dy$.
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How did he obtain: $df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$ ?

What do $df, dx$ and $dy$ mean to you? After considering that, maybe your problem is actually obvious? If not, then at least you have a much more concrete question to work on. — Arthur, Dec 30 '18 at 08:11
The Chain rule is based on the principle of composition of functions. We have a function $f(x,y)$ and in turn $x$ and $y$ are function of a parameter $t$. What we have is : $f(x(t), y(t))$. Thus, in order to compute $\dfrac {df}{dt}$ we have to apply the rule. — Mauro ALLEGRANZA, Dec 30 '18 at 08:20
More or less the same thing was asked just a few hours ago: https://math.stackexchange.com/questions/3056517/how-total-derivative-of-a-function-works-and-what-is-the-derivation-of-the-formu — Hans Lundmark, Dec 30 '18 at 08:43

How do we obtain $df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$ from $f(x,y)$?

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