Problems with implicit differentiation

[giant016]

In a problem where I need to use implicit diff. to find the slope of a line such as:

y^2 + x^2 = 9


2y (dy/dx) + 2x = 0


dy/dx = -y/x

Where does the (dy/dx) after the 2y (in the second part) come from? I've already differentiated y^2, x^2, and 9. Why isn't it just 2y + 2x = 0? I've looked all over, and in every problem an extra (dy/dx) or two just seem to pop up out of nowhere. Thanks!

 

[JonF]

Think of Y as a function in X. Then it's just the chain rule.

For example if y=e^x+x^3
y' = e^x + 3x^2

y^2 + x^2 = 9 is

(e^x+x^3)^2 + x^2 - 9 = 0
when we plug in y


differentiating this we get
2(e^x+x^3)(e^x + 3x^2) + 2x = 0

or 2y*(dy/dx) +2x = 0

when you do implicit differentiation you are doing it for a general y function of x instead of something specific (like as in our case of y=e^x+x^3)