The correct option is C 3x2√x+x3+ln62√x
If we closely watch the given function, it is the product of two different functions i.e., (x3+ln6) and √x. So don't you think that we can solve this by applying product rule for derivatives? The rule says that,
dd(x)[f(x)×g(x)]=dd(x)(f(x))×g(x))+ddx(g(x))×f(x))So, dd(x)[(x3+ln6)(√x)]=dd(x)(x3+ln6)×(√x)+dd(x)(√x)×(x3+ln6)=(3x2+0)(√x)+(x3+ln6)(12x−1/2)=3x2√x+x3+ln62√x
Which is the option C.
By the way this question can also be solved using addition rule for differentiation if you write the above function as,
x2√x+ln6√x
i.e. x5/2+ln6√x