I would like to find d y d x , where we have the implicit equation ( 2 x + y ) 4 + 3 x 2 y = y 3 .I just can't seem to understand the problem and which rule of calculus to use. Could someone please further elaborate?

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I would like to find                     d            y                      d            x      , where we have the implicit equation  (  2  x  +  y      )    4    +  3      x    2    y  =      y    3    .I just can&#039;t seem to understand the problem and which rule of calculus to use. Could someone please further elaborate?

Favatc6 · Accepted Answer

Just use the chain rule, the power rule, and the product rule and differentiate both sides of the equation.  4  (  2  x  +  y      )    3    (  2  +      y    ′    )  +  3      x    2        y    ′    +  6  x  y  =  3      y    2        y    ′        y    ′    (  4  (  2  x  +  y      )    3    +  3      x    2    −  3      y    2    )  =  −  6  x  y  −  8  (  2  x  +  y      )    3        y    ′    =            −      6      x      y      −      8      (      2      x      +      y              )        3                    4      (      2      x      +      y              )        3            +      3              x        2            −      3              y        2

ghettoking6q · Accepted Answer

If we differentiate the identity we get  2  ⋅  d  x  +  d  y  +  6  x  y  ⋅  d  x  +  3      x    2    ⋅  d  y  =  3      y    2    ⋅  d  yRearranging we get      (    2    +    6    x    y    )    ⋅  d  x  =      (    3          y      2        −    3          x      2        −    1    )    ⋅  d  yAnd we deduce                    d        y                    d        x              =                    2        +        6        x        y                    3                  y          2                −        3                  x          2                −        1            This is valid of course outside the hyperbola defined by       x    2    −      y    2    =  −            1      3

I would like to find dydx, where we have the implicit equation (2x+y)^4+3x^2y=y^3.

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