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

In summary, the problem is to find the equation of the tangent line to a curve with the equation x^(2/3) + y^(2/3) = a^(2/3) at the point (a, 0). The solution involves using the power rule and chain rule for implicit differentiation. The variable a is treated as a constant, so da/dx = 0. The solution may involve simplifying the expression to get a positive exponent in the numerator.
  • #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!
 
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  • #2
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
 

1. How do you differentiate implicitly with three variables?

Differentiating implicitly with three variables involves taking the derivative of an equation that has three variables (x, y, and a) with respect to one of the variables while treating the other two variables as constants.

2. What is the purpose of differentiating implicitly with three variables?

Differentiating implicitly with three variables allows us to find the rate of change of one variable with respect to another variable, even if the equation involves multiple variables.

3. What is the general process for differentiating implicitly with three variables?

The general process for differentiating implicitly with three variables involves taking the derivative of both sides of the equation with respect to the variable of interest, using the chain rule and product rule as necessary. The goal is to isolate the derivative of the variable of interest on one side of the equation.

4. How do you handle constants when differentiating implicitly with three variables?

When differentiating implicitly with three variables, constants are treated as just that – constants. This means that they can be factored out of the derivative and do not need to be differentiated.

5. Can you give an example of differentiating implicitly with three variables?

One example of differentiating implicitly with three variables is finding the derivative of the equation x^2 + y^2 = a^2 with respect to x. This would involve taking the derivative of both sides, using the chain rule to differentiate the y^2 term, and then solving for dy/dx.

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