# When to Implicit , When Not To?

1. Jun 22, 2012

### magorium

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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).

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 ?

Last edited: Jun 22, 2012
2. Jun 23, 2012

### HallsofIvy

Staff Emeritus
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.