1.

Differentiate each of the following w.r.t. x : (i) `sin (logx), x gt0` (ii) `log(logx), xgt1`

Answer» (i) Let `y=sin(logx)`.
Putting `logx=t,` we get
`y=sin t and t=logx`
`rArr(dy)/(dx)=cost and (dt)/(dx)=(1)/(x)`
`rArr(dy)/(dx)=((dy)/(dt)xx(dt)/(dx))=(costxx(1)/(x))=cos(logx)xx(1)/(x)=(cos(logx))/(x)`.
Hence, `(d)/(dx){sin(logx)}=(cos(logx))/(x).`
(ii) Let `y=log(logx)`.
Putting `logx=t,` we get
`y=logt and t=log x`
`rArr(dy)/(dx)=(1)/(t)and (dt)/(dx)=(1)/(x)`
`rArr(dy)/(dx)=((dy)/(dt)xx(dt)/(dx))=((1)/(t)xx(1)/(x))=((1)/(logx)xx(1)/(x))=(1)/((xlogx))`.
`therefore(d)/(dx){log(logx)}=(1)/((xlogx)).`


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