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How do you differentiate implicitly with three variables (x,y,a)?

  1. Nov 13, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the equation of the tangent line to the following curve at the indicated point:
    x^(2/3) + y^(2/3) = a^(2/3) at (a, 0)

    2. Relevant equations

    Power rule
    Chain rule

    3. The attempt at a solution

    (2/3)x^(-1/3) + [(2/3)y^(-1/3)](dy/dx) = [(2/3)a^(-1/3)](da/dx)

    Okay, from here I am stuck. I have no problem doing implicit differentiation when the only variables are x and y, but I have no idea what to do with that da/dx. The problem didn't tell me to treat a as a constant, so I assume I have to treat it as a variable (and when I tried treating it as a constant, I ended up solving for dy/dx and got 0 on the bottom, so it won't work anyways). I'm also a little confused about finding the derivative at (a, 0)... if a is not supposed to be a constant, how does this work?

    Basically, how do I find the derivative here when I have da/dx to worry about as well as dy/dx? Scratching my head. I don't even know where to begin.

    Any help would be MUCH appreciated! Thank you!
     
  2. jcsd
  3. Nov 13, 2011 #2

    LCKurtz

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    Science Advisor
    Homework Helper
    Gold Member

    a is obviously a constant, so da/dx = 0. Check your arithmetic; I think you will get 0 in the numerator, not the denominator if you express the answer with positive exponents.

    But there is a catch to this problem. Plot the graph using, for example, a = 1.
     
  4. Nov 14, 2011 #3
    Thank you so much, LCKurtz! My problem was that I didn't express the derivative with positive exponents. I have the right answer now. Thanks!!!!! :D
     
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