$\displaystyle e^y+6-e^-1 =3x^2 + 3y^2$
on the first term:
$\displaystyle e^y$
That uses chain rule? or does it remain the same?
The point was- the derivative with respect to what variable?
The derivative of $\displaystyle e^y$ with respect to y is $\displaystyle e^y$.
The derivative of $\displaystyle e^y$ with respect to x is $\displaystyle e^y \frac{dy}{dx}$.