Using implicit differentiation, we have that the derivative of 4 x 2 y − 3 / y = 0 is d y d x = − 8 x y 3 4 x 2 y 2 + 3 .But, if we multiply both sides of the original equation by y, we have the following equation 4 x 2 y 2 − 3 = 0 which is seemingly equivalent for y ≠ 0. Taking the derivative of this new function yields − y / x What is causing the difference here. Is it a rule of implicit differentiation or because the 0 in the original equation that allows us to rewrite it in such a form? I am new to implicit differentiation and any help is appreciated.

Question

Using implicit differentiation, we have that the derivative of   4      x    2    y  −  3      /    y  =  0 is            d      y              d      x        =  −            8      x              y        3                    4              x        2                    y        2            +      3        .But, if we multiply both sides of the original equation by y, we have the following equation   4      x    2        y    2    −  3  =  0 which is seemingly equivalent for   y  ≠  0. Taking the derivative of this new function yields   −  y      /    x What is causing the difference here. Is it a rule of implicit differentiation or because the 0 in the original equation that allows us to rewrite it in such a form? I am new to implicit differentiation and any help is appreciated.

na1p1a2pafr · Accepted Answer

Notice that if you take your equation   4      x    2        y    2    −  3  =  0 then you know that                    (1)                    4                  x          2                          y          2                                    =        3                            (2)                              y          2                                    =                  3                      4                          x              2                                      .            Now, taking the equation            d      y              d      x        =  −            8      x              y        3                    4              x        2                    y        2            +      3      and using (1) and (2) we get            d      y              d      x        =            −      8      x      y              (                  3                      4                          x              2                                      )              6    =  −            24      x      y              24              x        2              =  −      y    x    .So, your two forms of the derivative are equivalent, given the original equation. It is frequently the case that when doing implicit differentiation you may be able to obtain several seemingly different formulations of the derivative, which are equivalent given the original equation. As we see here, sometime one is much simpler than the other.

Using implicit differentiation, we have that the derivative of 4x^2y−3/y=0 is (dy)/(dx)=(−8xy^3)/(4x^2y^2+3).

Answered question

Answer & Explanation

New Questions in Calculus 2