What is total derivative?

Question

For some reason I can't for the life of me understand total derivative of multivariable function. I understand partial derivatives, you let one variable change and keep the others fixed, but total derivative doesn't make sense. By definition it's the best linear approximation of the function at a given point. So is it a linear transformation? So for example given the function $f(x,y)=x^2+y^2$ I can take the partial derivatives separately and get the function $f(x,y)=2x+2y.$ But if I have understood correctly that is different from the total derivative at point $(x,y).$ Could someone please explain it to me very carefully and simply. Any answers appreciated :)

Answer 1

At least in the special case of $f:\Bbb{R}^n\to \Bbb{R}~;~ f:\mathbf{x}\mapsto f(\mathbf{x})$, the total derivative of $f$ w.r.t an arbitrary variable $u$ is $$\frac{\mathrm{d}f}{\mathrm{d}u}=\sum_{i=1}^n \frac{\partial f}{\partial x_i}\frac{\mathrm{d}x_i}{\mathrm{d}u}$$ This is a straightforward formula, but I can provide some intuition behind it if you want. @KCd 's comment above briefly addresses the more general $\Bbb{R}^n\to\Bbb{R}^m$ case.