I recently attempted a question on implicit differentiation twice. I differentiated using one method in the first attempt and then another method in the second attempt but they do not correspond when I plug in values for the variables x and y. Please look at the two methods and tell me what I am doing wrong.
In the first attempt I just took the first derivative of each side of the equation with respect to x. In the second attempt, I took the ln of both sides of the equation first and then found the derivatives of either side of the equation.
Errata: The answer for the first attempt is y' = y(y - e^(x/y)) / (y ^ 2 - xe^(x/y))
On the second to last line of your first attempt, you had $y'=\frac{y(y-e^\frac{x}{y}y)}{y^2-e^{\frac{x}{y}}x}$ where actually it should be $y'=\frac{y(y-e^\frac{x}{y})}{y^2-e^{\frac{x}{y}}x}$. And then if you substitute $e^{\frac{x}{y}}$ with $x-y$, you will get $y'=\frac{y(y-(x-y))}{y^2-(x-y)x}=\frac{2y^2-xy}{y^2-x^2+xy}$, which is the same as the result from your second approach.