So I'm trying to differentiate an equation using implicit differentiation.I start with e x / y = 7 x − yNow the left side of the eqn is where I'm having trouble.I tried to use differentiation rules for exponentials, but this is incorrect.Here's what I tried though: ( e x ) 1 / y ln ⁡ ( e x ) y ′ = 7 − y ′ simplfied to: x ( e x ) 1 / y ⋅ y ′ = R S

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So I&#039;m trying to differentiate an equation using implicit differentiation.I start with       e          x              /            y        =  7  x  −  yNow the left side of the eqn is where I&#039;m having trouble.I tried to use differentiation rules for exponentials, but this is incorrect.Here&#039;s what I tried though:  (      e    x        )          1              /            y        ln  ⁡  (      e    x    )      y    ′    =  7  −      y    ′  simplfied to:  x  (      e    x        )          1              /            y        ⋅      y    ′    =  R  S

Marshall Mcpherson · Accepted Answer

I think the key thing to remember is:   y is a function of   x, so   x      y          −      1       is a function of   x, call it   u  (  x  ):   u  (  x  )  =  x      y          −      1      . Now the derivative of       e          u      (      x      )       is       e          u      (      x      )            u    ′    (  x  ), so we need       u    ′    (  x  )  =  (  x      y          −      1            )    ′    =      y          −      1        −  x      y          −      2            y    ′  . Thus      e          x              y                  −          1                      (      y          −      1        −  x      y          −      2            y    ′    )  =  7  −      y    ′  an equation which is linear in       y    ′  , for which we can solve using some simple algebra:  (  1  −      e          x              y                  −          1                      x      y          −      2        )      y    ′    =  7  −      e          x              y                  −          1                          y          −      1      or      y    ′    =            (      7      −              e                  x                      y                          −              1                                                  y                  −          1                    )              (      1      −              e                  x                      y                          −              1                                          x              y                  −          2                    )        =            (      7              y        2            −      y              e                  x                      y                          −              1                                          )              (              y        2            −      x              e                  x                      y                          −              1                                          )      which is about as far as we can go without knowing   y  (  x  ). Of course it should be remembeblack that, in deriving this formula, we have assumed that   y  (  x  )  ≠  0 is a differentiable function of   x.

So I'm trying to differentiate an equation using implicit differentiation. I start with e:(x/y)=7x−y Now the left side of the eqn is where I'm having trouble.

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