Question

For implicit differentiation, we assume that y is a function of x: we write y(x) to remind ourselves of this. However, for the circle x^2 + y^2 = 1 , it is not true that y is a function of x. Since y = +/- sqrt(1 - x^2) , there are actually (at least) two functions of x defined implicitly. Explain why this is not really a contradiction; that is, explain exactly what we are assuming when we do implicit differentiation.

Answer 1

are you there?

Answer 2

Did you try to differentiate it? If you do, you might get to the point where you can answer the question as it's desired.

Answer 3

no i havent

Answer 4

how do i do that?

Answer 5

It's a regular differentiating principle. As in the description, you watch y as a function of x, so you write y(x). If you do that however, you differentiate y(x) just as if it were a function of x. So

y(x)^2 differentiated will become 2y(x)*y'(x) by the chain rule.

Answer 6

ok

Answer 7

And there you already see why it's not a contradiction, because if you solve for the derivative y'(x) then you still have a function of y(x) in your denominator. You can solve for y(x) and again get two solutions. Just as in the regular case.

Answer 8

okay

Answer 9

so how would this not be a contradiction?

Answer 10

@precal @Mertsj @electrokid can u help?