Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Are implicit and partial differentiations related?

  1. Jan 8, 2012 #1
    I was working two different but superficially related problems, and noticed that if I did something that is generally not allowed, the results were connected by a negative sign. My questions are whether this will always turn out this way, and if so, why.
    The two problems were

    (A) implicit differentiation: given f(x, y(x)) = f(x,y) =(x^3)(y^2) = c for a constant c, then dy/dx = -3y/2x
    (B) partial differentiation: given f(x,y) =z=(x^3)(y^2) , then δf/δx = 3(x^2)(y^2) & δf/δy =2(x^3)y so doing something that is not allowed, (δf/δx)/(δf/δy) = δy/δx = 3y/2x.


    Coincidence, or can this be generalized (that dy/dx = the negative of working with the partial derivatives in this way), and the sloppiness justified? [My own impression is that it shouldn't be, but perhaps there is some better link between implicit and partial differentiations that someone can point out.]
     
  2. jcsd
  3. Jan 8, 2012 #2

    jedishrfu

    Staff: Mentor

    For partial differentiation the x and y are independent variables, but for the example you've shown y is dependent on x.

    Why not try another function like f(x,y)=sin(x^2 + y^2) and see if it holds up?
     
  4. Jan 8, 2012 #3
    right you are, jedishrfu. Thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Are implicit and partial differentiations related?
  1. Implicit Differentiation (Replies: 18)

  2. Implicit Differentiation (Replies: 12)

Loading...