Implicit differentiation question...

Question

anonymous · Answer

I think I understand why but I feel like I need an explanation.  When we have an equation such as $$x^2+y^2=25$$why do we use implicit differentiation vs. just solving for y and treating it like a normal function?  Again, I'm pretty sure I know the answer... just looking for confirmation.

amistre64 · Answer

whats your thought?

anonymous · Answer

As soon as I clicked "post" I knew I opened myself up for that one!  :)

anonymous · Answer

I think it's because you'd end up taking a sqrt which can give more than one value.

anonymous · Answer

But probably not now that I think about it because you do implicit differentiation with other types of equations as well.

amistre64 · Answer

your y' will still produce more than 1 value.  there are just cases that trying to solve for y at the start is a difficult task; if it is y' that we are seeking then we should just go straight for the juggular

anonymous · Answer

So it's just two methods of getting to the same place?

amistre64 · Answer

implicit has some theory behind it that allows us to conclude that solving explicitly for y is not a requirement.

anonymous · Answer

So I notice that when you take the derivative of f(x) you have no real "y" values that you're messing with.  When you use implicit differentiation you have both x and y values that you're having to deal with.

amistre64 · Answer

assume for the moment:

x = t;  dx/dt = 1

y = t^2;  dy/dt = 2t

what is dy/dx ?  well, y can be constructed as y=x^2 giving us 2x, but then we fail to see the thoery involved

amistre64 · Answer

on a different train of thought:

if I were to ask you what the derivative of 3x^2 was, what would you tell me

anonymous · Answer

6x

amistre64 · Answer

and I would say that you are in error ... why is that answer in error?

anonymous · Answer

oh, because it needs to be the derivative of something w/ respect to something else?  like $$\frac{ dy }{dx }$$

amistre64 · Answer

correct

amistre64 · Answer

we can implicly say:  the derivative of 3x^2 = 6x x'

amistre64 · Answer

if we then know further that this is with respect to x,  then x' = dx/dx = 1

anonymous · Answer

gotcha there... so I answered your question "implicitly"...

anonymous · Answer

dy/dx = -(∂F/∂x) / (∂F/∂y) = (-2x)/(2y) = -x/y

amistre64 · Answer

for some reason we condition the start of calculus to rely upon a wrt.x format that it becomes an assumption that overlooks the mechanics

anonymous · Answer

ok, thanks to the both of you for your time!

amistre64 · Answer

youre welcome