Let G ( x , y ) = x 2 y 4 − 3 x 4 y.(i) Find the first partial derivatives G x and G y .(ii) Using (i) above, find d y d x .(iii) If G ( x , y ) = 0, confirm your answer in part (ii) above, finding d y d x using implicit differentiation.

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Let   G  (  x  ,  y  )  =      x    2        y    4    −  3      x    4    y.(i) Find the first partial derivatives       G    x   and       G    y  .(ii) Using (i) above, find             d      y              d      x      .(iii) If   G  (  x  ,  y  )  =  0, confirm your answer in part (ii) above, finding             d      y              d      x       using implicit differentiation.

kidoceanoe · Accepted Answer

The chain rule says:  d  G  =            ∂      G              ∂      x        d  x  +            ∂      G              ∂      y        d  y  .If the point   (  x  ,  y  ) moves along a level set of   G, then we have   d  G  =  0. Hence            d      y              d      x        =            −      ∂      G              /            ∂      x                                −                    ∂      G              /            ∂      y        .  =  −            2      x              y        4            −      12              x        3            y                      x        2            4              y        3            −      3              x        4            and then we can cancel an   x.Now let&#039;s try implicit differentiation:      x    2        y    4    −  3      x    4    y  =  0.  2  x      y    4    +      x    2    4      y    3              d      y              d      x        −  12      x    3    y  −  3      x    4              d      y              d      x        =  0.Push the two terms not involving the derivative to the other side; then pull out the common factor, which is the derivative; then divide both sides by the other factor. We get            d      y              d      x        =            12              x        3            y      −      2      x              y        4                            x        2            4              y        3            −      3              x        4            and it&#039;s the same thing.

Let G(x,y)=x^2y^4−3x^4y. (i) Find the first partial derivatives G_x and G_y. (ii) Using (i) above, find dy/dx. (iii) If G(x,y)=0, confirm your answer in part (ii) above, finding dy/dx using implicit differentiation.

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