Question

Find the Derivative y’ implicitly.

Answer 1

`e^(x^2y) - e^y = e^y(e^(x^2) - 1) = x`

We should ISOLATE x

`e^y= (x)/(e^(x^2) - 1)`

Find the Natural Log of Both Sides to Make the Left Side "y"

`y = ln((x)/(e^(x^2)-1))`

Now, FIND THE DERIVATIVE Using Chain Rule!!!

`y' = (1)/((x)/(e^(x^2)-1)) * ((1(e^(x^2)-1) - x(2x*e^(x^2)))/((e^(x^2)-1)^2)= {(e^(x^2)-1)/(x)}* ((1(e^(x^2)-1) - x(2x*e^(x^2)))/((e^(x^2)-1)^2) = {(1)/(x)}* ((1(e^(x^2)-1) - x(2x*e^(x^2)))/((e^(x^2)-1)) = {(1)/(x)}* ((e^(x^2)-1 - 2x^2e^(x^2)))/((e^(x^2)-1))`