Question

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. $2 \cot x^{2}-4 x^{2} \,cosec\,{}^{2} x^{2}$
• C. $2 x \cot x^{2}$
• D. $-2 \,cosec\,{}^{2} x$

Answer 1

Answer: (D)

Given, $f(x)=\sin x, g(x)=x^{2}$ and $h(x)=\log _{e} x$
Also, $F(x)=($ hogof $)(x)=($ hog $)(\sin x)=h(\sin x)^{2}$
$\Rightarrow F ( x )=2 \log \sin x$
On differentiating, we get: $F '( x )=2 \cot x$
Again differentiating, we get: $F ''( x )=-2 \,cosec\,{}^{2} X$