So my problem is that I have to use implicit differentiation to find the derivative of F ( x , y ) = e x y − x when F ( x , y ) = 10 and the equation of the tangent at the point ( 1 , log ⁡ ( 11 ) ). So I tried to solve this using two ways:The first way I used was rearranging the equation to 10 + x = e x y and then using ln to simply into ln ⁡ ( 10 + x ) = x y before using implicit differentiation. The result is d y d x = 1 10 x + x 2 − y x and after substituting the point, I get approximately −0.950 as the slope.The second way I used was just differentiating it straight away rather than rearranging and I get d y d x = 1 e x y − y x .But when I sub the point in I get approximately −0.688 as the slope. So I'm not sure if I've done something wrong when getting the derivatives or am I not allowed to rearrange the equation?

Question

So my problem is that I have to use implicit differentiation to find the derivative of   F  (  x  ,  y  )  =      e          x      y        −  x when   F  (  x  ,  y  )  =  10 and the equation of the tangent at the point   (  1  ,  log  ⁡  (  11  )  ). So I tried to solve this using two ways:The first way I used was rearranging the equation to   10  +  x  =      e          x      y       and then using   ln to simply into   ln  ⁡  (  10  +  x  )  =  x  y before using implicit differentiation. The result is            d      y              d      x        =      1          10      x      +              x        2              −      y    x  and after substituting the point, I get approximately −0.950 as the slope.The second way I used was just differentiating it straight away rather than rearranging and I get            d      y              d      x        =                    1                  e                      x            y                              −      y        x    .But when I sub the point in I get approximately −0.688 as the slope. So I&#039;m not sure if I&#039;ve done something wrong when getting the derivatives or am I not allowed to rearrange the equation?

Danica Ray · Accepted Answer

Careful, I edited what I thought was a typo, but it turned out to be the source of your error, so I reversed the edit.  F  (  x  ,  y  )  =  10 when   (  x  ,  y  )  =  (  1  ,  ln  ⁡  (  11  )  ), not when   (  x  ,  y  )  =  (  1  ,  log  ⁡  (  11  )  ), where I (and your calculator) are using   log  ⁡  (  11  ) to mean the base-10 logarithm, not the base-e logarithm. Both your derivatives are correct, and when you plug in   (  1  ,  ln  ⁡  (  11  )  ) they both give you            d      y              d      x        ≈  −  2.307.