TITLE: Implicit Differentiation
QUESTION [8 upvotes]: I was just wondering where the y'/(dy/dx) in implicit differentiation comes from.
$$
x^2 + y^2 = 25
$$
$$
(d/dx) x^2 + (d/dy) y^2 **(dy/dx)** = 25 (d/dx)
$$
$$
2x + 2y (dy/dx) = 0
$$
$$
(dy/dx) = -x/y
$$
Where does the bold part come from? Wikipedia says it's a byproduct of the chain rule, but it's just not clicking for me.

REPLY [9 votes]: Isaac and Ryan have already answered your question in words.  Now, in symbols, the chain rule gives:
$$\frac{d(y^2)}{dx} = \frac{d(y^2)}{dy}\frac{dy}{dx} = 2y\frac{dy}{dx}$$