What does dy/dx really mean in the context of implicit differentiationThe problem is that I can't relate it back to the limit definition. If you can't write y explicitly as a function of x, often y approaches multiple values as x approach some value, or at some x, y has multiple values hence the limit does not exist and therefore the derivative will not exist

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What does dy/dx really mean in the context of implicit differentiationThe problem is that I can&#039;t relate it back to the limit definition. If you can&#039;t write y explicitly as a function of x, often y approaches multiple values as x approach some value, or at some x, y has multiple values hence the limit does not exist and therefore the derivative will not exist

Spielgutq1 · Accepted Answer

For implicit differentiation you will have an equation that -- at least locally -- is satisfied by the points on a curve through the plane.Imagine that we find a parameterization of the curve such that the set in question is  {  (      f    x    (  t  )  ,      f    y    (  t  )  )  ∣  t  ∈  [  0  ,  1  ]  }and otherwise &quot;sufficiently well-behaved&quot;. Then at the point   (      f    x    (      t    0    )  ,      f    y    (      t    0    )  ) the implicit derivative ought to be the limit      lim          h      →      0                          f        y            (              t        0            +      h      )      −              f        y            (              t        0            )                      f        x            (              t        0            +      h      )      −              f        x            (              t        0            )

What does dy/dx really mean in the context of implicit differentiation

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