Find d y d x for x 2 = x − y x + y I have solved this in two ways.First, I multiplicated the whole equation by x + y and then I calculated the implicit derivative. I got the following solution: 1 − 3 x 2 − 2 x y x 2 + 1 So far so good. When I calculated the implicit derivative of the original expression using the quotient rule though, I got a different solution, i.e.: − x ( y + x ) 2 − y x Can anyone explain to me why I get different solutions ?

Question

Find             d      y              d      x       for       x    2    =            x      −      y              x      +      y      I have solved this in two ways.First, I multiplicated the whole equation by   x  +  y and then I calculated the implicit derivative. I got the following solution:            1      −      3              x        2            −      2      x      y                      x        2            +      1      So far so good. When I calculated the implicit derivative of the original expression using the quotient rule though, I got a different solution, i.e.:  −            x      (      y      +      x              )        2            −      y        x  Can anyone explain to me why I get different solutions ?

Steppkelk · Accepted Answer

The two solutions that you found are equal. So where is no contradiction.Note that solving         x    2    =            x      −      y              x      +      y         for   y gives     y  =  x            1      −              x        2                    1      +              x        2            Your first solution :            1      −      3              x        2            −      2      x      y                      x        2            +      1        =            1      −      3              x        2            −      2      x              (        x                              1            −                          x              2                                            1            +                          x              2                                      )                            x        2            +      1        =            1      −      4              x        2            −              x        4                    (      1      +              x        2                    )        2            Your second solution :  −  (  x  +  y      )    2    +      y    x    =  −            (      x      +      x                        1          −                      x            2                                    1          +                      x            2                              )        2    +      1    x        (    x                  1        −                  x          2                            1        +                  x          2                      )    =            1      −      4              x        2            −              x        4                    (      1      +              x        2                    )        2            Thus            1      −      3              x        2            −      2      x      y                      x        2            +      1        =  −  (  x  +  y      )    2    +      y    x

Find (dy)/(dx) for x^2=(x−y)/(x+y).

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