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Question
Differentiate the following function w.r.t.x. : `((x+1)(x-1))/(("e"^x+1))`
Solution
Let y = `((x + 1)(x - 1))/(("e"^x + 1))`
∴ y = `(x^2 - 1)/(("e"^x + 1))`
Differentiating w.r.t. x, we get
`dy/dx=d/dx((x^2 - 1)/("e"^x + 1))`
= `(("e"^x + 1)d/dx(x^2 - 1) - (x^2 - 1)d/dx("e"^x + 1))/("e"^x + 1)^2`
= `(("e"^x + 1)(2x) - (x^2 - 1)("e"^x + 0))/("e"^x + 1)^2`
= `(2x"e"^x + 2x - x^2"e"^x + "e"^x)/("e"^x + 1)^2`
= `(2x"e"^x + "e"^x - x^2"e"^x + 2x)/("e"^x + 1)^2`
= `("e"^x(2x + 1 - x^2) + 2x)/("e"^x + 1)^2`
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