# What is total derivative?

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 :)

• See here math.stackexchange.com/questions/3754445/…. For the gradient, have a look at math.stackexchange.com/questions/3159730/…. Commented Jul 20, 2020 at 17:26
• This paper has a nice review of multivariate differentiation in section $5$. Commented Jul 20, 2020 at 17:26
• The total derivative is a linear transformation. If $f \colon \mathbf R^n \to \mathbf R^m$ is described componentwise as $f(\mathbf x) = (f_1(\mathbf x), \ldots, f_m(\mathbf x))$, for $\mathbf x$ in $\mathbf R^n$, then the total derivative of $f$ at $\mathbf x$ is the $m \times n$ matrix $(\partial f_i/\partial x_j)$ where the partial derivatives are computed at $\mathbf x$. For example, if $f \colon \mathbf R^2 \to \mathbf R$ by $f(x,y) = x^2+y^2$ then the total derivative of $f$ at $(x,y)$ is the $1 \times 2$ matrix $(2x \ \ 2y)$.
– KCd
Commented Jul 20, 2020 at 17:42
• I’m voting to close this question because OP has in a typical fashion not reacted on comments or answers. Commented Sep 7, 2023 at 9:33

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.