# [Multivariable Calculus] Implicit Function Theorem

1. Oct 13, 2016

### David Donald

I am having trouble doing this problem from my textbook... and have
no idea how to doit.

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

I am having trouble doing this problem from my textbook...

Show that the equation x + y - z + cos(xyz) = 0 can be solved for z = g(x,y) near the origin. Find dg/dx and dg/dy

(dg/dx and dg/dy are partial derivatives)
2. Relevant equations
Implicit function theorem.

3. The attempt at a solution
I tried computing dg/dx and dg/dy like it told me but
I think that isn't what its asking..

2. Oct 13, 2016

### Staff: Mentor

I think that the trick here is to recognize that if x and y are both close to 0, then xyz is also close to zero, so what can you say about cos(xyz)?

3. Oct 13, 2016

### Ray Vickson

You are given an equation of the form $f(x,y,z)=0$ and an initial point $(0,0,z_0)$ that satisfies it (where I will let you figure out the value of $z_0$). The implicit function theorem states that for certain conditions on the derivatives of $f$ in a neighborhood of $(0,0,z_0)$, the equation is solvable for $z$ in terms of $(x,y)$, near $(0,0)$. Does your given $f$ satisfies those conditions? Does the theorem apply to your function?