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Differentiate (log x)^(cos x) with respe...

Differentiate `(log x)^(cos x)` with respect to x.

Text Solution

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`"Let " y = ("log "x)^("cos "x)`
`rArr "log " y = "log "{("log "x)^("cos "x)}`
`= "cos "x * "log "("log "x)`
Diffenentiate both sides w.r.t.x
`(1)/(y) (dy)/(dx) = "cos "x * (d)/(dx)"log" ("log " x) + "log"("log " x)(d)/(dx)"cos"x`
`rArr (dy)/(dx) = y {("cos"x)/("log"x)*(d)/(dx)("log"x)+"log"("log" x)(-"sin"x)}`
`rArr (dy)/(dx) =("log" x)^("cos"x){("cos"x)/(x*"log"x)-"sin"x * "log" x ("log"x)}`
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