Q. Let $f(x) = \sin{x,g(x) = x^{2}}\ and\ h(x) = \log_{e}{x.If\ F(x) = (hogof)(x),\ \ then\ F^{''}(x)}\$is equal to
A) $a\ {cosec}^{3}\ x$
B) $2\cot{x^{2} - 4x^{2}\ {cosec}^{2}\ x^{2}}$
C) $2x\cot x^{2}$
D) $- 2\ {cosec}^{2}\ x$
Explanation
(d)
$\lbrack hog\rbrack(x) = h\left( x^{2} \right) = 2\ log_{e}\ x$
$\Rightarrow \ (hogof)(x) = hog(\sin{x) = 2\log_{e}{\sin x}}$
$\Rightarrow F(x) = \ 2\log_{e}{\sin x}\$
$\Rightarrow F^{'}(x) = 2\cot x\$
$\Rightarrow F^{''}(x) = \ - 2\ cosec^{2}\ x$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.