Partial derivatives chain rule

[sid9221]

Suppose we have a function $$V(x,y)=x^2 + axy + y^2$$
how do we write
$$\frac{dV}{dt}$$

For instance if $$V(x,y)=x^2 + y^2$$, then $$\frac{dV}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}$$

So, is the solution

$$\frac{dV}{dt} = 2x \frac{dx}{dt} + ay\frac{dx}{dt} + ax\frac{dy}{dt} + 2y \frac{dy}{dt}$$

 

[LCKurtz]

Yes.

 

[ChrisVer]

Just for generality, whenever you have a function $F(x(t),y(t),...,z(t), t)$ (it is a function of some functions of t, and also depends explicitly on t, for example:
$F= x(t)+y(t)^{2}+...+lnz(t) + (t^{3}-t^{2})$
and you want to find its derivative, you have:
$\frac{dF}{dt}= \frac{\partial F}{\partial x}|_{y,...,z=const}\frac{dx}{dt}+\frac{\partial F}{\partial y}|_{x,...,z=const}\frac{dy}{dt}+...+\frac{\partial F}{\partial z}|_{x,y,...=const}\frac{dz}{dt} + \frac{\partial F}{\partial t}|_{x,y,...,z=const}$