Question

Let $f\left(x\right)=\sin x,g\left(x\right)=x^2,h\left(x\right)=\log x$,If $F\left(x\right)=h\left\{f\right(g\left(x\right)\left)\right\}$, then $\frac{d^{2}F}{d{x}^2}=$

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

Answer 1

The correct option is B $2\cot x^2-4x^2\cos e{c}^2{x}^2$

$F\left(x\right)=log\sin x^2$
$\frac{dF\left(x\right)}{dx}=\frac{2x\cos \left(x^2\right)}{\sin x^2}$

$\frac{d^{2}F\left(x\right)}{d{x}^2}=\frac{(2\cos x^2-4x^2\sin x^2)\sin x^2-(2x\cos x^2{)}^2}{{\sin }^2{x}^2}$

$\frac{d^{2}F\left(x\right)}{d{x}^2}=2\cot x^2-4x^{2}cosec2x^2$