Question

Differentiate `y=(sin(x))^(log(x))`?

Answer 1

`(dy)/(dx)=0.4343(sinx)^(logx)(sinx/x+cotxlnx)`

Explanation:

As `y=(sin(x))^(log(x))`

we have `lny=ln(sin(x))^(log(x))=logxln(sinx)=lnx/ln10ln(sinx)`

Now as `lny=lnx/ln10ln(sinx)`

`1/y(dy)/(dx)=1/ln10(1/xln(sinx)+lnx*1/sinx*cosx)`

= `1/2.3026(sinx/x+cotxlnx)`

or `(dy)/(dx)=1/2.3026(sinx/x+cotxlnx)xx(sinx)^(logx)`

= `0.4343(sinx)^(logx)(sinx/x+cotxlnx)`