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Suppose y + ln(1 + x) = 0. Which of the following statements is true?

1. e^y = x*y' + 1
2. y' = -1 / (x + 1)
3. y' + e^y = 0
4. y' = e^y

Correct answer: e^y = x*y' + 1

Solution

Since y = -ln(1+x), we get y' = -1/(1+x) and e^y = e^(-ln(1+x)) = 1/(1+x). Checking option A: x*y' + 1 = -x/(1+x) + 1 = 1/(1+x) = e^y. Options B and C are also individually correct; A is the combined identity stated as a single relation. All of A, B, C hold; D does not.

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