I've been stuck on a certain implicit differentiation problem that I've tried several times now. x 2 x + y = y 2 + 6I know to take the derivatives of both sides and got: ( x + y ) 2 x − ( 1 − d y d x ) x 2 ( x + y ) 2 = 2 y d y d x I blackuced that to get: 2 x 2 + 2 x y − x 2 − x 2 ∗ d y / d x = ( 2 y ∗ d y / d x ) ( x + y ) 2 I then divided both sides by (2y*dy/dx) and multiplied each side by the reciprocals of the first three terms of the left side. Then I factoblack dy/dx out of the left side and multiplied by the reciprocal of what was left to get dy/dx by itself. I ended up with: d y / d x = ( 2 y ( x + y ) 2 ) / ( 4 x 7 y )but this answer was wrong. I only have one more attempt on my online homework and I can't figure out where I went wrong. Please help!

Question

I&#039;ve been stuck on a certain implicit differentiation problem that I&#039;ve tried several times now.            x      2              x      +      y        =      y    2    +  6I know to take the derivatives of both sides and got:            (      x      +      y      )      2      x      −              (        1        −                              d            y                                d            x                          )                    x        2                    (      x      +      y              )        2              =  2  y            d      y              d      x      I blackuced that to get:  2      x    2    +  2  x  y  −      x    2    −      x    2    ∗  d  y      /    d  x  =  (  2  y  ∗  d  y      /    d  x  )  (  x  +  y      )    2  I then divided both sides by (2y*dy/dx) and multiplied each side by the reciprocals of the first three terms of the left side. Then I factoblack dy/dx out of the left side and multiplied by the reciprocal of what was left to get dy/dx by itself. I ended up with:  d  y      /    d  x  =  (  2  y  (  x  +  y      )    2    )      /    (  4      x    7    y  )but this answer was wrong. I only have one more attempt on my online homework and I can&#039;t figure out where I went wrong. Please help!

Jazlene Dickson · Accepted Answer

An idea to avoid the cumbersome and annoying quotient rule: multiply by the common denominator            x      2              x      +      y        =      y    2    +  6    ⟹    x      y    2    +  6  x  +      y    3    +  6  y  −      x    2    =  0    ⟹        y    2    +  2  x  y      y    ′    +  6  +  3      y    2        y    ′    +  6      y    ′    −  2  x  =  0    ⟹    (  2  x  y  +  3      y    2    +  6  )      y    ′    =  2  x  −      y    2    −  6    ⟹        y    ′    =            2      x      −              y        2            −      6              2      x      y      +      3              y        2            +      6      If nevertheless you want to use the quotient rule:            2      x      (      x      +      y      )      −              x        2                    (      x      +      y              )        2              −            x      2              (      x      +      y              )        2                  y    ′    =  2  y      y    ′      ⟹    (  2  y  (  x  +  y      )    2    +      x    2    )      y    ′    =      x    2    +  2  x  y    ⟹        y    ′    =                    x        2            +      2      x      y              2      y      (      x      +      y              )        2            +              x        2            Now, how come both expressions we got are equal?. Well, for one we can use, for example, that                                                                 x            2                                                                            x          +          y                      =      y    2    +  6   , so                    x        2            +      2      x      y              2      y      (      x      +      y              )        2            +              x        2              =            2      x      −              y        2            −      6              2      x      y      +      3              y        2            +      6        =            2      x      −                        x          2                          x          +          y                            2      x      y      +      2              y        2            +                        x          2                          x          +          y                        ⟺                      x        2            +      2      x      y              2      y      (      x      +      y              )        2            +              x        2              =                    x        2            +      2      x      y              2              x        2            y      +      4      x              y        2            +      2              y        3            +              x        2                ⟺                      x        2            +      2      x      y              2              x        2            y      +      4      x              y        2            +      2              y        3            +              x        2              =                    x        2            +      2      x      y              2              x        2            y      +      4      x              y        2            +      2              y        3            +              x        2                      ✓

I've been stuck on a certain implicit differentiation problem that I've tried several times now. (x^2)/(x+y)=y^2+6

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