I don't really understand what to do for implicit differentiation. For example, how do I go about doing this:
Find dy/dx by implicit differentiation. 4x2 + 4xy - y2 = 1
Implicit differentiation is really just the chain rule in disguise.
If you have a function f(y) and take y = y(x) then you've got f(y(x)). Take the x derivative and you get f'(y(x)) y'(x) by the chain rule. This is exactly the same thing as saying
So let's take your problem:
Take the derivative:
Now solve for
-Dan
for implicit differentiation, you just find the derivative of each variable, taking account for what you are differentiating and with respect to what.
if you differentiate an x-term you attach dx/dx to it (since you took the derivative of the variable x with respect to x), we usually don't write it though, because derivative notations can function as fractions and so dx/dx cancels to become 1
similarly, when differentiating a y-term, we attach dy/dx to it, since we took the derivative of y with respect to x. we often write y' when it is understood what we are differentiating with respect to
here goes:
differentiating implicitly, we get:
now solve for , which is
EDIT: Beaten by "the man." And he gave a better explanation too!