How do I simplify If e^2y−e^(y2−y)=x4−x2 using implicit differentiation?

Answer 1

Implicit differentiation is nothing more than differential forms where we treat the infinitesimal quantites as "itty bitty" numbers.

So...

e^(2y) has a differential form of e^(2y) * 2 * dy

-e^(y^2 - y) has a differential form of -e^(y^2 - y) * (2y - 1) * dy

x^4 - x^2 has a differential form of (4x^3 - 2x) *dx

Putting this together gives...

[ 2e^(2y) - (2y -1)*e^(y^2 - y) ] * dy = (4x^3 - 2x) * dx

Therefore dy/dx = (4x^3 - 2x)/[ 2e^(2y) - (2y -1)*e^(y^2 - y) ]

If you had a specific point, you would be able to determine the slope at that point...