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

  • Thread starter Antebellum
  • Start date
  • #1

Homework Statement



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)

Homework Equations



Power rule
Chain rule

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!
 

Answers and Replies

  • #2
LCKurtz
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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.
 
  • #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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