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. $a\cos e{c}^{3}x$
• B. $a\cot x^2-4x^2\cos e{c}^2{x}^2$
• C. $2x\cot x^2$
• D. $-2\cos e{c}^{2}x$
• E. $4\cos e{c}^{2}x$

Answer 1

Answer: (D)

Given, $f(x)=\sin x,g(x)=x^2$ and $h(x)={\log }_{e}x$
Also, $f(x)=(hogof)(x)$
Now, $[hog](x)=2{\log }_{e}x$
$\Rightarrow$ $(hogof)(x)=2{\log }_{e}x$
$\Rightarrow$ $f(x)=2{\log }_e\sin x$
On differentiating w.r.t. $x,$
we get $f(x)=2\cot x$
Again differentiating,
we get $f(x)=-2\cos e{c}^{2}x$