Hi Folks,

if we have an implicit function $\displaystyle z=x^2-4y^2+5xy-6=0$ eqn1

then we can find the standard derivative $\displaystyle \frac{dz}{dx}=2x-8y\frac{dy}{dx}+5x\frac{dy}{dx}+5y$ eqn 2 and

the partial derivative $\displaystyle \frac{\partial z}{\partial x}=2x+5y$ eqn3

However, I am trying to grasp the idea of having a standard/total derivation and partial derivation of functions as above yet we dont have standard or partial integration (ie the reverse). How is that?

If we integrate back eqn2 wrt to x we get eqn1 with a constant. This constant can be determined if BC's or IC's are known (ie 6) etc...but

if we integrate eqn3 wrt x we can get eqn1 with a constant also but we lose the $\displaystyle -4y^2$ term. How does one get this term back?

I dont think I understand this fully. Can anyone shed some light?

Thanks

Bugatti79