Does anyone have any idea how to do the Implicit Differentiation in Calculus?

Question

It is really confusing me! For example, how would you do x/(x+y)? (x divided x+y). Or if you just have any advice, I'm all ears!

sorry, but yes, the question is with respect to x.

Answer 1

Implicit differentiation is really jsut an exercise in using the chain rule. For instance, if you had e^(x^2), its derivative would be e^(x^2) * 2x, since d(x^2)/dx = 2x. It's the same thing with implicit, except that you don't actually know what dy/dx is yet, so you just write dy/dx. So for my example, d(e^y)/dx = e^y dy/dx. Now, onto your problem:

d(x/(x+y))/dx = ((x+y) - (1+dy/dx)x)/(x+y)², or being simplified, (y-x dy/dx)/(x+y)²

Of course, this assumes you are differentiating with respect to x. If you are differentiating w.r.t. a third variable (e.g. t), then you would have to use the chain rule on x as well. So your derivative would be:

d(x/(x+y))/dt = (dx/dt (x+y) - (dx/dt + dy/dt)x)/(x+y)²

Answer 2

Basically the only difference between implicit diff. and reg. differentiation is how instead of the derivative of a number (X) being 1, the derivative of Y is DY.
To do this problem you must do quotient rule. When you take the derivative of Y, turn it into DY instead of a 1, then solve for DY.