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Question
Find the derivative of the following w. r. t.x. : `(3e^x-2)/(3e^x+2)`
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Solution
Let y = `(3"e"^x - 2)/(3"e"^x + 2)`
Differentiating w.r.t. x, we get
`dy/dx = d/dx((3"e"^x - 2)/(3"e"^x + 2))`
=`((3"e"^x + 2) d/dx(3"e"^x - 2) - (3"e"^x - 2) d/dx(3"e"^x + 2))/((3"e"^x + 2)^2)`
= `((3"e"^x + 2)(d/dx(3"e"^x) - d/dx(2)) - (3"e"^x - 2)(d/dx(3"e"^x) + d/dx(2)))/((3"e"^x + 2)^2)`
= `((3"e"^x + 2)(3"e"^x - 0) - (3"e"^x - 2)(3"e"^x + 0))/((3"e"^x + 2)^2)`
= `(3"e"^x(3"e"^x + 2) - 3"e"^x(3"e"^x - 2))/((3"e"^x + 2)^2)`
= `(3"e"^x(3"e"^x + 2 - 3"e"^x + 2))/(3"e"^x + 2)^2`
= `(3"e"^x(4))/(3"e"^x + 2)^2`
= `(12"e"^x)/(3"e"^x + 2)^2`
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