The next step would be to solve the equation for y' for finding the slope of the tangent - but you need to look again at your differentiating. You've got the implict bit, using the chain rule, but you need the product rule outside of that...
use implicit differentiation to find the equation of the tangent line to the curve given by equation xe^y = y-1 at the point where y=1
i know i'm supposed to implicit the equation and what i get is:
xe^y*y' = 1+y'
this is where i'm stuck, and not sure how to proceed with what i have
Picture on its way, but basically x and e^y are the two parts of the product which need the product rule. So you get x' times e^y plus x times (e^y)'
(e^y)' is the bit you've already applied the chain rule to (correctly) but then just multiplied by x (instead of applying the needed product rule)
Good on the left, check the right
The pic, just in case it helps...
... according to taste, where
... is the chain rule, wrapped here inside the legs-uncrossed version of...
... the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
Don't integrate - balloontegrate!