Notation for a derivative

Question

I am interested if there is notation for a derivative that is in between a total derivative and partial derivative.

The total derivative of $f(t,x,y)$ with respect to $t$ is $$ \frac{df}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt} $$ while the partial derivative of $f(t,x,y)$ with respect to $t$, holds $x$ and $y$ constant, and is $\frac{\partial f}{\partial t}$.

I am interested in an intermediary derivative that, say only holds $x$ constant, and is equal to $$ \frac{df}{dt}=\frac{\partial f}{\partial t}+0+\frac{\partial f}{\partial y}\frac{dy}{dt} $$ Is there any notation I can use for this kind of derivative?

Answer 1

I am writing this answer since I believe that too many beginners overrate the concept of total derivative. In mathematics we differentiate functions, not "variables". When the OP wrote $$ \frac{df}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}, $$ he used the symbol $f$ with at least two different meanings.

In the left-hand side $f$ is understood as a function of a single (real) variable. In the right-hand side, $f$ is thought of as a function of three (real) variables. This identity is what we call an abuse of notation.

Moreover, the so-called total derivative is essentially a ghost: it is a notation for lazy people who do not want to be rigorous and write

let $x=x(t)$, $y=y(t)$ be two (differentiable) functions, and let $f \colon \mathbb{R}^3 \to \mathbb{R}$ be a differentiable function. If $g(t)=f(t,x(t),y(t))$, then $$g'(t) = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} x'(t)+\frac{\partial f}{\partial y}y'(t).$$

The total derivative is just the chain rule for lazy people, so to say. If we want to fix one variable, say $x$, and differentiate $h(t)=f(t,x,y(t))$, we'd rather state it clearly to avoid any misunderstanding.