Undergrad Chain rule in a multi-variable function

[Ananthan9470]

Suppose you have a parameterized muli-varied function of the from $F[x(t),y(t),\dot{x}(t),\dot{y}(t)]$ and asked to find $\frac{dF}{dt}$, is this the correct expression according to chain rule? I am confused because of the derivative terms involved.

$\frac{dF}{dt}=\frac{\partial F}{\partial x} \frac{dx}{dt} + \frac{\partial F}{\partial y} \frac{dy}{dt}$

Or similar terms containing $\dot{x}(t)$ etc should also be included or it is something else altogether?

 

[Mark44]

Suppose you have a parameterized muli-varied function of the from $F[x(t),y(t),\dot{x}(t),\dot{y}(t)]$ and asked to find $\frac{dF}{dt}$, is this the correct expression according to chain rule? I am confused because of the derivative terms involved.

$\frac{dF}{dt}=\frac{\partial F}{\partial x} \frac{dx}{dt} + \frac{\partial F}{\partial y} \frac{dy}{dt}$

Or similar terms containing $\dot{x}(t)$ etc should also be included or it is something else altogether?

If the function had parameters x, y, z, and w, the total derivative would have four terms, with the last two being $\frac{\partial F}{\partial z} \frac{dz}{dt} + \frac{\partial F}{\partial w} \frac{dw}{dt}$. I believe that the derivative you're trying to find needs similar terms, with the partials being with respect to $\dot{x}$ and $\dot{y}$.