Differentiate `(xcosx)^x`with respect to `xdot`

1.	Differentiate `(xcosx)^x`with respect to `xdot`
Answer» Correct Answer - `(x cos x)^(x)[( log x +1) +{ log cos x- x cot x}]` Let `y = (x cos x)^(x).` Taking logarithm on both sides, we get `log y = log (x cos x)^(x)` `=x log ( x cos x )` `=x log x+ x log cos x` Differentiating both sides with respect to x, we get `(1)/(y)(dy)/(dx)=xcdot(1)/(x)+ log x+log cos x+xcdot ((-sin x)/(cos x))` `" or "(dy)/(dx)=( x cos x)^(x)[(log x +1)+{ log cos x - x cot x}]`

Discussion

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If `f(theta) = sin(tan^(-1)(sintheta/sqrt(cos2theta)))`, where `-pi/4 lt theta lt pi/4,` then the value of `d/(d(tantheta)) f(theta)` is