Question

If e^x + e^y = e^x+y , prove that `("d"y)/("d"x) = -"e"^(y - x)`

Answer 1

Given that e^x + e^y = e^x+y.

Differentiating both sides w.r.t. x, we have

`"e"^x + "e"^y ("d"y)/("d"x) = "e"^(x + y) (1 + ("d"y)/("d"x))`

or `("e"^y - "e"^(x + y)) ("d"y)/("d"x) = "e"^(x + y) - "e"^x`

Which implies that `("d"y)/("d"x) = ("e"^(x + y) - "e"^x)/("e"^y - "e"^(x + y))`

= `("e"^x + "e"^y - "e"^x)/("e"^y - "e"^x - "e"^y)`

= –e^y–x.