Given is F ( x , y ) = y e 3 x − 2 x 2 = 0I was asked to calculate y ′ using implicit differentiation.I know that y ′ = − F x F y = − ∂ F ∂ x ∂ F ∂ y .So I obtained: y ′ ( x ) = − 3 y e 3 x − 4 x e 3 x = − 3 y e 3 x + 4 x e 3 x .But, in the solution manual I found another approach: y = f ( x )i.e. f ( x ) e 3 x − 2 x 2 = 0, We use the product rule ⇒ f ′ ( x ) e 3 x + f ( x ) 3 e 3 x − 4 x = 0 f ′ ( x ) e 3 x + f ( x ) 3 e 3 x = 4 x f ′ ( x ) e 3 x = 4 x − ( f ( x ) 3 e 3 x ) f ′ ( x ) = 4 x − ( 3 e 3 x f ( x ) e 3 x And finally f ′ ( x ) = y ′ ( x ) = 4 x − 3 e 3 x y e 3 x = − 3 y e 3 x + 4 x e 3 x So, the result is the same.Are both approaches valid? Is there any difference between them? Which one would you recommend to use?

Question

Given is   F  (  x  ,  y  )  =  y      e          3      x        −  2      x    2    =  0I was asked to calculate       y    ′   using implicit differentiation.I know that       y    ′    =  −            F      x              F      y        =  −                              ∂                F                              ∂                x                                      ∂                F                              ∂                y              .So I obtained:       y    ′    (  x  )  =  −            3      y              e                  3          x                    −      4      x              e              3        x              =            −      3      y              e                  3          x                    +      4      x              e              3        x              .But, in the solution manual I found another approach:  y  =  f  (  x  )i.e.   f  (  x  )      e          3      x        −  2      x    2    =  0, We use the product rule  ⇒      f    ′    (  x  )      e          3      x        +  f  (  x  )  3      e          3      x        −  4  x  =  0      f    ′    (  x  )      e          3      x        +  f  (  x  )  3      e          3      x        =  4  x      f    ′    (  x  )      e          3      x        =  4  x  −  (  f  (  x  )  3      e          3      x        )      f    ′    (  x  )  =            4      x      −      (      3              e                  3          x                    f      (      x      )              e              3        x            And finally      f    ′    (  x  )  =      y    ′    (  x  )  =            4      x      −      3              e                  3          x                    y              e              3        x              =            −      3      y              e                  3          x                    +      4      x              e              3        x            So, the result is the same.Are both approaches valid? Is there any difference between them? Which one would you recommend to use?

Pradellalo · Accepted Answer

The two approaches are indeed equivalent. In the general case, you have an equation  F  (  x  ,  f  (  x  )  )  =  0where   y  =  f  (  x  ).Differentiating both sides with respect to   x, you get            ∂      F              ∂      x        (  x  ,  f  (  x  )  )  +      f    ′    (  x  )            ∂      F              ∂      y        (  x  ,  f  (  x  )  )  =  0  ,and solving for       f    ′    (  x  ), you have      f    ′    (  x  )  =  −                              ∂          F          (          x          ,          f          (          x          )          )                          ∂          x                                      ∂          F          (          x          ,          f          (          x          )          )                          ∂          y                      ,which is precisely the implicit function theorem formula that you quoted originally.

Given is F(x,y)=ye^(3x)−2x^2=0 I was asked to calculate y′ using implicit differentiation.

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