The total derivative as I understand it would be a linear map along the lines on the wikipedia page. https://en.wikipedia.org/wiki/Total_derivative
However in some instances it looks alot like a chain rule or people use the chain rule to compute it.
I dont see why we would assume that the variables are functions of some common parameter $t$ when we talk about total derivative.
Why are these thing being mixed everywhere? Are they related somehow?
Best Answer
It is just a special case for the chain rule with
$$f=f(t,x(t),y(t),...)$$
More precisely, total derivative is a special case of composition $f\circ g:\mathbb{R}\to\mathbb{R} $ with $f:\mathbb{R^n}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R^n}$.
Indeed, in a more general case with
$$f=f(x(u,v),y(u,v),z(u,v)....)$$
we can apply chain rule to evaluate partial derivatives of $f$ with respect to $u$ and $v$
$$\frac{\partial f}{\partial u} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial u}+ \frac{\partial f}{\partial z} \cdot \frac{\partial z}{\partial u}+...$$
$$\frac{\partial f}{\partial v} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial v} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial v}+ \frac{\partial f}{\partial z} \cdot \frac{\partial z}{\partial v}+...$$
In this special case, since $x,y,...$ are function of one variable $t$, we have
$$\frac{d f}{d t} =\frac{\partial f}{\partial t} = \frac{\partial f}{\partial t} \cdot \frac{\partial t}{\partial t} + \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial t}+ \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial t}+...=\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} \cdot \frac{dx}{dt}+ \frac{\partial f}{\partial y} \cdot \frac{dy}{dt}+...$$
we denote this special case as total derivative of $f$ with respect to $t$.