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Question
Differentiate the following w.r.t.x. :
y = `(5"e"^x - 4)/(3"e"^x - 2)`
Solution
y = `(5"e"^x - 4)/(3"e"^x - 2)`
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = "d"/("d"x) ((5"e"^x - 4)/(3"e"^x - 2))`
`("d"y)/("d"x) = ((3"e"^x - 2) "d"/("d"x) (5"e"^x - 4) - (5"e"^x - 4) "d"/("d"x) (3"e"^x - 2))/(3"e"^x - 2)^2`
= `((3"e"^x - 2)(5 "d"/("d"x) "e"^x - "d"/("d"x) 4) - (5"e"^x - 4)(3 "d"/("d"x) "e"^x - "d"/("d"x) 2))/(3"e"^x - 2)^2`
= `((3"e"^x - 2)(5"e"^x - 0) - (5"e"^x - 4)(3"e"^x - 0))/(3"e"^x - 2)^2`
= `(15("e"^x)^2 - 10"e"^x - 15("e"^x)^2 + 12"e"^x)/(3"e"^x - 2)^2`
= `(2"e"^x)/(3"e"^x - 2)^2`
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