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