We covered implicit differentiation recently, and I'm relatively clear on it, but I'd like to clarify a few things.
I'll first explain the example and then explain my questions.
(a) If x^2 + y^2 = 25, find dy/dx
(b) Find an equation of the tangent line to the circle x^2 +y^2 = 25 at the point (3, 4).
(a) Questions: I'm not quite clear on why the derivative of y^2 = 2y * y' rather than simply 2y. I understand that it's because y is a function of x, but don't understand why multiplying by y' is the solution.
Also, I don't quite understand why, when they solved the equation 2x+2yy' = 0 for y', they said that y' = -(x/y). I'm much more lost on this aspect of part (a). Is it simply always this way?
(b) Questions: The only thing I'm unclear with for part (b) is why y' (-3/4) is used as the slope of the tangent line as opposed to x'.
Thank you very much, this forum is a great resource!