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?

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    $\begingroup$ 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... $\endgroup$ – Siminore Apr 6 '15 at 15:20
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    $\begingroup$ You really shouldn't use a notation to suggest that $\frac{dx}{dt} = 0$. You should just say it explicitly. $\endgroup$ – DanielV Apr 6 '15 at 15:26
  • $\begingroup$ @Siminore I was hoping there would be some notation like $\frac{d f}{d t}_{\textrm{hold $x$ constant}}=\ldots$ $\endgroup$ – user103828 Apr 6 '15 at 15:39
  • $\begingroup$ @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 $\frac{\partial f}{\partial t}+\frac{\partial f}{\partial y}\frac{d y}{d t}$" $\endgroup$ – user103828 Apr 6 '15 at 15:39
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    $\begingroup$ My post here may help you understand the concept of partial derivative: math.stackexchange.com/questions/1091097/… $\endgroup$ – 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.

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.

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  • $\begingroup$ I agree that it may be that beginners misuse \ abuse the notation but I think there are places where the total derivative is useful and places where the partial derivative is useful. The example I took comes from Wikipedia en.wikipedia.org/wiki/Total_derivative $\endgroup$ – user103828 Apr 6 '15 at 16:28
  • $\begingroup$ Partial derivatives are meaningful when you want to differentiate a function that is defined on a cross product $F \colon X_1 \times X_2 \to Y$. $\endgroup$ – Siminore Apr 6 '15 at 17:01

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