The correct option is C 2sinxcosx
All the formulae mentioned for the derivative of standard functions hold true only when the variable is x itself, not for any other function of x. To make it more clear let’s take lnx as the function. Its derivative with respect to x as we know will be
1x, but the derivative of lnsinx with respect to x will not be 1sinx. Rather it will be 1sinx multiplied by d(sinx)dx. Yes the derivative of lnsinx with respect to sinx will be 1sinx but not with respect to x
In this question the function is of the form of xn. The only difference is that it has sinx at the place of x. So we will use chain rule.
.dydx=dyd(sinx)×d(sinx)dx
dyd(sinx)=2sinx
d(sinx)dx=cosx
So dydx=2sinxcosx
Also you can write the function as sinx.sinx and try to solve it using product rule.