1. The problem statement, all variables and given/known data For the given function z to demonstrate the equality: As you see I show the solution provided by the book, but I have some questions on this. I don't understand how the partial derivative of z respect to x or y has been calculated. Do you think this is correct? I think this is a giant errata, I guess the function z is not given implicitly and it simply is: z = ln ( x ^2 + y^2) The partial derivatives are calculated normally: ∂z/∂x= 2 * x/(x^2 + y^2) Similar for y, and with this it is straighforward to demonstrate the equality. What do you think? There are two options: 1.- Or the statement and solution of the given problem is correct---> In that case I don't understand anything. Could you explain how to get the partial derivatives? 2.- Or there is a giant errata, z is not given implicitly and the calculation is easy. And forgeting this problem I was wondering in case I found an equation with z given implicitly like: z^2 = x * z + y * z^3 How would we differenciate this equation? As we have many variables we should choose which are maintained constant and which are changing. Suppose we differenciate this expression considering x is changing, y is constant but z obviously changes, due to the changes in x. The receipt is changing x for x+dx, z changes to z+dz and y doesn't change at all. I get: (z+dz)2 - z2 = ( (x+dx) * (z+dz) + y * (z+dz)3 ) - (x * z + y * z3) After neglecting diferentials of order two and three I get: dz = dx * (z dx / 2x - x -3 y z^2) But this differential arose because there was a change on x, so I should call it dzx, then I should do the same calculation for dzy and the total differential of the function z should be: dz = dzx + dz y Is this correct?