text: # Partial Derivatives Proof
  
  1. Sep 13, 2011
  
  ### KingBigness
  
  1. The problem statement, all variables and given/known data
  Suppose that the equation F(x,y,z) = 0 implicitly defines each of the three variables x, y and z as functions of the other two: z = f(x,y), y = g(x,z), z = h(y,z). If F is differentiable and Fx, Fy and Fz are all nonzero, show that
  
  $\frac{∂z}{∂x}$ $\frac{∂x}{∂y}$ $\frac{∂y}{∂z}$ = -1
  
  3. The attempt at a solution
  
  I have been scribbling away at this for a little while now and can't see it.
  I assume that I need to prove that each partial derivative = -1, thus -1*-1*-1=-1
  
  Don't want you guys to simply give me the answer but any tips on where to start would be greatly appreciated =D
  
  Last edited: Sep 13, 2011
  2. Sep 13, 2011
  
  ### Dick
  
  Try a simple example. Say, x+2y+3z=0. You can then solve for f, g and h explicitly and show your product is (-1). And you'll also see that not all of the partials need to be (-1). That should be a good hint to start.
  
  3. Sep 13, 2011
  
  ### KingBigness
  
  That worked...but how can I solve for all cases?
  
  4. Sep 14, 2011
  
  ### KingBigness
  
  I tried ax+by+cz=0 and solved it and ended up with -1....but this still doesn't solve it for all cases does it?
  
  5. Sep 14, 2011
  
  ### Dick
  
  It sort of does. Since F is differentiable and invertible it can be approximated by a linear function well enough. If you want to be more formal then use the chain rule (read all of the following d's as partial derivatives). E.g. let's try finding dx/dy in terms of F. When you are taking a partial derivative you need to specify what variable is being held constant. In this case it must be z. Now the chain rule for partial derivatives tells you (dF/dx)*(dx/dy)+(dF/dy)*(dy/dy)+(dF/dz)*(dz/dy)=0, right? If you hold z constant, can you solve that for dx/dy in terms of partial derivatives of F? Repeat that for dz/dx and dy/dz and multiply the results together.
  
  Last edited: Sep 14, 2011
  6. Sep 14, 2011
  
  ### KingBigness
  
  Thank you so much for that.
  I had tried using the chain rule, but I mixed it up =( I only started teaching myself partial derivatives 2 days ago so I'll give my self a break for stuffing up =P
  
  Thank you again, I'll give that a shot and see how it goes
  
  7. Sep 14, 2011
  
  ### Dick
  
  That's ok. This one is a little subtle. I had to ponder it a while.
  
  8. Sep 14, 2011
  
  ### KingBigness
  
  haha I'm glad you had trouble with it to, makes me feel less dumb =P
explanation: The document is a forum post discussing a math problem involving partial derivatives. The user "KingBigness" presents a problem and asks for tips on how to solve it. Another user, "Dick," provides guidance and hints to help solve the problem. The post contains a question and an ongoing discussion toward finding a solution.
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distilabel_metadata.role: system
distilabel_metadata.content: # Partial Derivatives Proof
  
  1. Sep 13, 2011
  
  ### KingBigness
  
  1. The problem statement, all variables and given/known data
  Suppose that the equation F(x,y,z) = 0 implicitly defines each of the three variables x, y and z as functions of the other two: z = f(x,y), y = g(x,z), z = h(y,z). If F is differentiable and Fx, Fy and Fz are all nonzero, show that
  
  $\frac{∂z}{∂x}$ $\frac{∂x}{∂y}$ $\frac{∂y}{∂z}$ = -1
  
  3. The attempt at a solution
  
  I have been scribbling away at this for a little while now and can't see it.
  I assume that I need to prove that each partial derivative = -1, thus -1*-1*-1=-1
  
  Don't want you guys to simply give me the answer but any tips on where to start would be greatly appreciated =D
  
  Last edited: Sep 13, 2011
  2. Sep 13, 2011
  
  ### Dick
  
  Try a simple example. Say, x+2y+3z=0. You can then solve for f, g and h explicitly and show your product is (-1). And you'll also see that not all of the partials need to be (-1). That should be a good hint to start.
  
  3. Sep 13, 2011
  
  ### KingBigness
  
  That worked...but how can I solve for all cases?
  
  4. Sep 14, 2011
  
  ### KingBigness
  
  I tried ax+by+cz=0 and solved it and ended up with -1....but this still doesn't solve it for all cases does it?
  
  5. Sep 14, 2011
  
  ### Dick
  
  It sort of does. Since F is differentiable and invertible it can be approximated by a linear function well enough. If you want to be more formal then use the chain rule (read all of the following d's as partial derivatives). E.g. let's try finding dx/dy in terms of F. When you are taking a partial derivative you need to specify what variable is being held constant. In this case it must be z. Now the chain rule for partial derivatives tells you (dF/dx)*(dx/dy)+(dF/dy)*(dy/dy)+(dF/dz)*(dz/dy)=0, right? If you hold z constant, can you solve that for dx/dy in terms of partial derivatives of F? Repeat that for dz/dx and dy/dz and multiply the results together.
  
  Last edited: Sep 14, 2011
  6. Sep 14, 2011
  
  ### KingBigness
  
  Thank you so much for that.
  I had tried using the chain rule, but I mixed it up =( I only started teaching myself partial derivatives 2 days ago so I'll give my self a break for stuffing up =P
  
  Thank you again, I'll give that a shot and see how it goes
  
  7. Sep 14, 2011
  
  ### Dick
  
  That's ok. This one is a little subtle. I had to ponder it a while.
  
  8. Sep 14, 2011
  
  ### KingBigness
  
  haha I'm glad you had trouble with it to, makes me feel less dumb =P
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a forum post discussing a math problem involving partial derivatives. The user "KingBigness" presents a problem and asks for tips on how to solve it. Another user, "Dick," provides guidance and hints to help solve the problem. The post contains a question and an ongoing discussion toward finding a solution.
  
  Label: question-and-answer