Implicit differentiation I want to differentiate 
$x^2 + y^2=1$ with respect to $x$.
The answer is $2x +2yy' = 0$.
Can some explain what is implicit differentiation and from where did $y'$ appear ? I can understand that $2x +2yy' = 0$ is a partial derivative but then it becomes multi calc not single.
This is in a chapter about chain rule so I assume there is some use of the chain rule here but I can't see any composite functions here. We can express y in terms of x but y is not composite.
P.S: I am NOT looking for HOW to solve the problem, I am looking for the WHY as stated above.

In this case we consider $ y $ as function of $ x $, so the differential of $ \frac{dy^{2}}{dx}=2y\frac{dy}{dx}$ where $ y^{'}=2y\frac{dy}{dx}$ thus
\begin{align}
\dfrac{d\left[ x^{2}+y^{2}\right]}{dx}  &= \dfrac{d\left[ x^{2}\right]}{dx}+ \dfrac{d\left[ y^{2}\right]}{dx}\\
     &= 2x + \frac{dy^{2}}{dx}=2x+2y\frac{dy}{dx} \\
\left[ x^{2}+y^{2}\right]^{'} &= 2x+ 2yy^{'}.
\end{align} 
