In Maple, I have a function $f(x(t),y(t),t)$ that I want to differentiate with respect to $t$. I know the command for partial derivative $\frac{\partial f}{\partial x}$,$\frac{\partial f}{\partial y}$,$\frac{\partial f}{\partial t}$, etc in Maple. But I can't find a way to find total derivative $\frac{df}{dt}$.
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2$\begingroup$ So just do it explicitly $\endgroup$– Maxim UmanskyCommented Nov 24, 2021 at 15:54
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$\begingroup$ Have you tried the diff from the Physics package? $\endgroup$– TyberiusCommented Nov 25, 2021 at 2:13
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1 Answer
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You have this function:
$$ f = f(x,y,t) $$
so:
$$ df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial t} dt $$
or:
$$ \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} $$
You said that, you are able to calculate $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, and $\frac{\partial f}{\partial t}$. So, just calculate them and I assume you know the relations of $x = x(t)$, $y = y(t)$ as well. So, everything is ready to find the $\frac{df}{dt}$ here.
answered Nov 24, 2021 at 17:17
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1$\begingroup$ Thanks, but I need a general symbol for this in MAPLE. I need to find total derivatives several times. Doing the manual process is not feasible. $\endgroup$– ilawidCommented Nov 24, 2021 at 20:25
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1$\begingroup$ @ilawid I worked with Maple many years ago, but I believe it should be easy to automate this process. Basically it's a function that you put your symbolical $f$, $x=x(t)$, and $y=y(t)$ into it and it just calculates $\partial_{x} f$, $\partial_{y} f$, $\partial_{t} f$, $\dot{x}$, and $\dot{y}$ and returns $\dot{f}$ based on that formula. I'm not sure why you think you need to do it manually. Note: at the end $\dot{f}$ might not be something really compact or even it's not guaranteed that you would be able to write it down based on only $t$ parameter. It just depends on explicit formula of $f$. $\endgroup$ Commented Nov 24, 2021 at 20:50