If $y=\sin ^{-1}\left(\frac{\log x^2}{1+(\log x)^2}\right)$, then $\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)_{\mathrm{at ~} x=1}=$

If $f(x)=\left\{\begin{array}{cc}\frac{a}{2}(x-|x|) & , \\ 0, & \text { for } x<0 \\ 0, & \text { for } x=0 \\ b x^2 \sin \left(\frac{1}{x}\right) & \text { for } x>0\end{array}\right.$

is continuous at $x=0$, then

The p.m.f. of a random variable X is given by

$$\begin{aligned} \mathrm{P}[\mathrm{X}=x] & =\frac{\binom{5}{x}}{2^5}, \text { if } x=0,1,2,3,4,5 \\ & =0, \text { otherwise } \end{aligned}$$

Then which of the following is not correct?

If $\mathrm{f}(x)=(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)$, then $f^{\prime}(1)=$