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Question
If ex + ey = ex+y , prove that `("d"y)/("d"x) = -"e"^(y - x)`
Solution
Given that ex + ey = ex+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)`
= –ey–x.
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