# Notation for a derivative

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?

• You could always write $$\frac{d}{dt}f(t,x,y(t)),$$ but you must say clearly that $x$ is a fixed number. You can take so many partial and total derivatives that is would be rather pointless to introduce another piece of notation... – Siminore Apr 6 '15 at 15:20
• You really shouldn't use a notation to suggest that $\frac{dx}{dt} = 0$. You should just say it explicitly. – DanielV Apr 6 '15 at 15:26
• @Siminore I was hoping there would be some notation like $\frac{d f}{d t}_{\textrm{hold$x$constant}}=\ldots$ – user103828 Apr 6 '15 at 15:39
• @DanielV from what I understand, you're suggesting something like, "... holding $x$ constant the total derivative of $f(t,x,y)$ wrt to $t$ is $\frac{\partial f}{\partial t}+\frac{\partial f}{\partial y}\frac{d y}{d t}$" – user103828 Apr 6 '15 at 15:39
• My post here may help you understand the concept of partial derivative: math.stackexchange.com/questions/1091097/… – DanielV Apr 7 '15 at 4:53

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.
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.
• Partial derivatives are meaningful when you want to differentiate a function that is defined on a cross product $F \colon X_1 \times X_2 \to Y$. – Siminore Apr 6 '15 at 17:01