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Question
Differentiate the following w.r.t. x : `[(tanx)^(tanx)]^(tanx) "at" x = pi/(4)`
Solution
Let y = `[(tanx)^(tanx)]^(tanx)`
∴ log y = `log[(tanx)^(tanx)]tanx`
= tanx. log(tanx)tanx
= tanx. tanx log(tan x)
= (tanx)2. log(tan x)
Differentiating both sides w.r.t. x, we get
`1/y."dy"/"dx" = "d"/"dx"[tanx)^2.log(tanx)]`
= `(tanx)^2."d"/"dx"(log tanx) + (log tanx)."d"/"dx"(tanx)^2`
= `(tanx)^2. xx 1/tanx."d"/"dx"(tanx) + (log tanx) xx 2tanx."d"/"dx"(tanx)`
= `(tanx)^2 xx 1/tanx.sec^2x + (log tanx) xx 2 tanxsec^2x`
∴ `"dy"/"dx" = y[(tanx)(sec^2x) + (logtanx)(2tanxsec^2x)]`
= [(tanx)tanx]tanx.(tanxsec2x)[1 + 2logtanx]
If x = `pi/(4)`, then
`"dy"/"dx" = [(tan pi/4)^(tan pi/4)]^(tan pi/4)(tan pi/4 sec^2 pi/4)[1 + 2log tan pi/4]`
= `[(1)^1]^1.[1(sqrt(2))^2][1 + 2log1]`
= 1 x 2 x 1 ...[∵ log 1 = 0]
= 2.
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