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Question
Differentiate the following:
y = `"e"^(3x)/(1 + "e"^x`
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Solution
y = `"e"^(3x)/(1 + "e"^x`
`("d"y)/("d"x) = ((1 + "e"^x) "e"^(3x) (3) - "e"^(3x) (0 + "e"^x))/(1 + "e"^x)^2`
`("d"y)/("d"x) = (3(1 + "e"^x) "e"^(3x) - "e"^(3x) * "e"^x)/(1 + "e"^x)^2`
`("d"y)/("d"x) = (3"e"^(3x) + 3"e"^x "e"^(3x) - "e"^x "e"^(3x))/(1 + "e"^x)^2`
= `(3"e"^(3x) + 2"e"^x "e"^(3x))/(1 + "e"^x)^2`
= `(3"e"^(3x) + 2"e"^(x + 3x))/(1 + "e"^x)^2`
`("d"y)/("d"x) = (3"e"^(3x) + 2"e"^(4x))/(1 + "e"^x)^2`
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