Let f(x)=sinx,g(x)=x2,h(x)=logex. If F(x)=(hogof)(x), then F''(x) is equal to
acosec3x
2cotx2-4x2cosec2x2
2cotx2
-2cosec2x
Explanation for the correct option.
Find the value of F''(x):
Given,
f(x)=sinx,g(x)=x2,h(x)=logex and
F(x)=h(g(f(x)))=h(g(sinx))=h(sin2x)=logesin2x
Now differentiate F(x) w. r. t. x.
F'(x)=1sin2x2sinxcosx=2cotx
Again differentiate F'(x) w.r.t. x.
F''(x)=-2cosec2x[∵ddx(cotx)=-cosec2x]
Hence the correct option is D.
Use the factor theorem to determine whether g(x) is a factor of f(x)
f(x)=22x2+5x+2;g(x)=x+2