So my problem is that I have to use implicit differentiation to find the derivative...

Faith Welch

Faith Welch

Answered

2022-07-17

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?

Answer & Explanation

Danica Ray

Danica Ray

Expert

2022-07-18Added 15 answers

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.

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