Find the third derivative of the
x^2 e^-y + y^2 e^-1 = 1
Sir/Maam, may I know how did you get to feel so easy answering math problems? Did you ever find it hard when you're a student?
Of course there were things I found hard when I was a student.
There are still things I find hard. Once you understand them, then they are easy, and you can move on to the next hard thing.
So, you want to find derivatives here. Use implicit differentiation. Just remember the chain rule and the product rule.
e^-1? I doubt it. I suspect you meant
x^2 e^-y + y^2 e^-x = 1
All those e^-stuff get in the way.
How about writing it as
x^2/e^y + y^2/e^x = 1
x^2 e^x + y^2 e^y = e^(x+y)
(x^2 + 2x) e^x + (y^2 + 2y)e^y y' = e^(x+y) (1+y')
Now collect terms to get
y' = (2x e^x - y^2 e^y)/(x^2 e^x - 2y e^y)
Now use the quotient rule to find y"
y" = {[2(x+1)e^x - (y^2+2y)e^y y'](x^2 e^x - 2y e^y) - (2x e^x - y^2 e^y)[(x^2+2x)e^x - 2(y+1) e^y y']}/(x^2 e^x - 2y e^y)^2
Now plug in y' and simplify, using Algebra I.
Hah! Good luck with that!
Third derivative? You'd better find some good online symbolic differentiator. Or, just type up some gibberish. Your teacher will not be able to tell whether it's right or not!