Answer
$$y'=-\frac{1}{2}$$
Work Step by Step
$$e^{x+2y}=1$$
Using implicit differentiation, we differentiate both sides of the equation with respect to $x$: $$e^{x+2y}\times(x+2y)'=0$$
$$e^{x+2y}\times(1+2y')=0$$
$$e^{x+2y}+2e^{x+2y}y'=0$$
Next, set all the elements having $y'$ to one side and the rest to the other side: $$2e^{x+2y}y'=-e^{x+2y}$$
Finally, calculate for $y'$: $$y'=-\frac{e^{x+2y}}{2e^{x+2y}}=-\frac{1}{2}$$
