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Question
Differentiate the following w.r.t.x:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
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Solution
Let y = `(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
= `(e^(2x) - 1/e^(2x))/(e^(2x) + 1/e^(2x))`
= `(e^(4x) - 1)/(e^(4x) + 1)`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"((e^(4x) - 1)/(e^(4x) + 1))`
= `((e^(4x) + 1)."d"/"dx"(e^(4x) - 1) - (e^(4x) - 1)."d"/"dx"(e^(4x) + 1))/(e^(4x) + 1)^2`
= `((e^(4x) + 1)[e^(4x)."d"/"dx"(4x) - 0] - (e^(4x) - 1)[e^(4x)."d"/"dx"(4x) + 0])/(e^(4x) + 1)^2`
= `((e^(4x) + 1).e^(4x) xx 4 - (e^(4x) - 1).e^(4x) xx 4)/(e^(4x) + 1)^2`
= `(4e^(4x)(e^(4x) + 1 - e^(4x) + 1))/(e^(4x) + 1)^2`
= `(4e^(4x)(cancel(e^(4x)) + 1 - cancel(e^(4x)) + 1))/(e^(4x) + 1)^2`
= `(4e^(4x)(1 + 1))/(e^(4x) + 1)^2`
= `(4e^(4x)(2))/(e^(4x) + 1)^2`
= `(8e^(4x))/(e^(4x) + 1)^2`.
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