How to find d y / d xfrom x 3 y 6 = ( x + y ) 9 using implicit differentiation? I tried solving but I ended up with solution that does not agree with my textbook answer. How can I get d y / d x = y / x

Question

How to find  d  y      /    d  xfrom      x    3        y    6    =  (  x  +  y      )    9  using implicit differentiation? I tried solving but I ended up with solution that does not agree with my textbook answer. How can I get  d  y      /    d  x  =  y      /    x

rmercierm7 · Accepted Answer

The answer does appear to be correct. Here&#039;s how to go about it:1) First, use implicit differentiation to get  3      x    2        y    6    +  6      x    3        y    5              d      y              d      x        =  9  (  x  +  y      )    8        (    1    +                  d        y                    d        x              )    .2) Solve for             d      y              d      x       to get            d      y              d      x        =            9      (      x      +      y              )        8            −      3              x        2                    y        6                    6              x        3                    y        5            −      9      (      x      +      y              )        8              .3) Multiply the numerator and denominator of the RHS by   (  x  +  y  ) to get            d      y              d      x        =            9      (      x      +      y              )        9            −      3              x        2                    y        6            (      x      +      y      )              6              x        3                    y        5            (      x      +      y      )      −      9      (      x      +      y              )        9              .4) Using the original formula for   (  x  +  y      )    9  , this becomes            d      y              d      x        =            9              x        3                    y        6            −      3              x        2                    y        6            (      x      +      y      )              6              x        3                    y        5            (      x      +      y      )      −      9              x        3                    y        6              .5) By simplifying, we get            d      y              d      x        =            6              x        3                    y        6            −      3              x        2                    y        7                    6              x        4                    y        5            −      3              x        3                    y        6              =            2      x      y      −              y        2                    2              x        2            −      x      y        .6) By factoring common factors, we get            d      y              d      x        =            y      (      2      x      −      y      )              x      (      2      x      −      y      )        =      y    x    .

babuliaam · Accepted Answer

First simplify your equation by taking the cube root of each side to get  x      y    2    =  (  x  +  y      )    3  Then take the derivative of both sides and do the usual process to find            d      y              d      x        =            3      (      x      +      y              )        2            −              y        2                    2      x      y      −      3      (      x      +      y              )        2            Then multiplying numerator and denominator by   x  +  y,            d      y              d      x        =            3      (      x      +      y              )        3            −              y        2            (      x      +      y      )              2      x      y      (      x      +      y      )      −      3      (      x      +      y              )        2              =            3      x              y        2            −      x              y        2            −              y        3                    2              x        2            y      +      2      x              y        2            −      3      x              y        2              =            2      x      y      −              y        3                    2              x        2            y      −      x              y        2              =                    y        2            (      2      −      y      )              x      y      (      2      −      y      )        =      y    x

How to find dy/dx from x^3y^6=(x+y)^9 using implicit differentiation?

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