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}