Calculating partial derivative of multivariable function

I want to calculate the derivative $\frac{d F}{d t}$ of a function $F(x(t),y(t),t)$. So i need to calculate $\frac{d F}{d t} = \frac{\partial F}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial t}$. Is this the right way?

Solution 1:

So i need to calculate $\frac{d F}{d t} = \frac{\partial F}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial t}$. Is this the right way?

Almost. $F(x(t),y(t),t)$ has three arguments, so applying the Chain Rule::$$\frac{d F}{d t} = \frac{\partial F}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial t}+\frac{\partial F}{\partial t}\frac{\partial t}{\partial t}$$

Where, of course, $\tfrac{\partial t}{\partial t}=1$.