When to Implicit , When Not To?

[magorium]

Homework Statement

We have a surface z^2 = 4x^2 + 2yx + 5y^2 , find the shortest distance to Origin.

Homework Equations

The Attempt at a Solution

My trouble is , i think z^2 - 4x^2 + 2yx + 4y^2 = 0 as a constraint to function L = x^2 + y^2 + z^2 (Square of distance formula. If distance is minimum , square of it should be too).

Now i will use grad(L) = Lambda*grad(surface)

But while finding the gradient of surface , i need derivatives respect to x , y and z as vector components. Now , while taking the derivative of constraint respect to x , should i use the implicit differentiaton ? My book just directly takes the differential of it respect to x , doesn't use the implicit diff.

When dealing with plane equations , multivariable calculus , when to use implicit diff and when not to ? How could i understand if z is a function of x and y or they all are three variables ?

 

[HallsofIvy]

When you take partial derivatives with respect to x, y, or z, you are, by definition, treating the other variables as constant. That is why you do not differentiate y and z, for example, with respect to x when taking the partial derivative with respect to x.