confusion on the equation of circle Let us consider the equation $x^2+y^2=a^2$.we all know that this is a equation of a circle.
clearly it is a differentiable function.
But for each value of $x$ we can get two different values of $y$.so the function is not well defined and hence not a function.Then how can I differentiate it?
can someone remove my confusion please.
 A: There are two ways you can think of this.  Obviously this does not define a function in itself, since for any $x$ there are two $y$'s we may choose.
By implicit differentiation we find: $\frac{dy}{dx} = -\frac{x}{y}$.
What this says is that given any function $y = f(x)$ that satisfies: $$x^2 + y^2 = a^2$$ Then the derivative of that function is $f'(x) = -x/f(x)$.  We know two functions that would satisfy this: $f_1(x) = \sqrt{a^2 - x^2}$ and $f_2(x) = -\sqrt{a^2 - x^2}$.  There are more than just those two functions, but those are the only ones that would be continuous from $[-a,a]$.  We say that the equation $x^2 + y^2 = a^2$ implicitly defines a function.
Another way to look at this is that the equation $x^2 + y^2 = a^2$ defines a curve in the plane.  It is a curve that is not the graph of any function, since it fails the vertical line test.  However, we can still talk about the slope of a tangent line on that curve.  That would be what we mean by the implicit derivative at a point.
A: $$x\space dx + y\space dy = 0\\
\frac{dy}{dx} = \frac{-x}{y}$$
