Hello. I know how to do implicit differentiation taught in calculus 1, but I'm confused by something regarding it.(adsbygoogle = window.adsbygoogle || []).push({});

Take the example:

y^{3}+y^{2}-5y-x^{2}=4

If we do implicit differentation we get:

3y^{2}(dy/dx)+2y(dy/dx)-5(dy/dx)-2x=0

dy/dx=2x/(3y^{2}+2y-5)

Now, it makes sense how to compute it, but I have troubles really understanding why it works. For example, if I graph the original function I find it fails the vertical line test (for example at -2) so I know it's not a function. But I only really understand what it means for a function to be differentiable. I guess what I mean is, when I think of something being differentiable, I think of a continuous function. What exactly is the original equation? How can I think of it as being continuous or differentiable or any such things? Can I use epsilon and delta definitions like I would normally to determine things like limits and continuity and differentiability?

I hope this question makes sense. It's just, I know how to do the implicit differentiation, I just have difficulty thinking of taking the derivative of anything other than an actual function, if that makes sense.

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# Implicit differentiation - major confusion

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