Question

Let $f\left(x\right)=\sin x$; $g\left(x\right)=x^2$ and $h\left(x\right)=\log x$. If $u\left(x\right)=h\left(f\left(g\left(x\right)\right)\right)$, then $\frac{d^{2}u}{d{x}^2}$ is

• A. $2{\cos }^{3}x$
• B. $2\cot x^2-4x^{2}cose{c}^2{x}^2$
• C. $2x\cot x^2$
• D. $-2cose{c}^{2}x$

Answer 1

Answer: (B)

$2\cot x^2-4x^{2}cose{c}^2{x}^2$

$u\left(x\right)=h\left(f\left(g\left(x\right)\right)\right)=h\left(f\left(x^2\right)\right)=h\left(\sin x^2\right)=\log \sin x^2$
Hence $u^{\prime }\left(x\right)=2x\cot x^2$ and $u^{\prime \prime }\left(x\right)=2\cot x^2-4x^{2}co{\sec }^2{x}^2$.