Let $\mathrm{y}\left(\mathrm{x}\right)=(1+\mathrm{x})(1+{\mathrm{x}}^2)(1+{\mathrm{x}}^4)(1+{\mathrm{x}}^8)(1+{\mathrm{x}}^{16})$ . Then ${\mathrm{y}}^{\text{'}}-{\mathrm{y}}^{\text{'}\text{'}}$  at $\mathrm{x}=-1$  is equal to:

$\mathrm{y}=(\mathrm{x}+1)({\mathrm{x}}^2+1)({\mathrm{x}}^4+1)({\mathrm{x}}^8+1)({\mathrm{x}}^{16}+1)$
Multiplying and dividing by (x – 1) we get
$\mathrm{y}=\frac{{\mathrm{x}}^{32}-1}{\mathrm{x}-1}$
at x = -1, y = 0
$\mathrm{y}(\mathrm{x}-1)={\mathrm{x}}^{32}-1$
Diff. on both side 
$\mathrm{y}\text{'}(\mathrm{x}-1)+\mathrm{y}=32{\mathrm{x}}^{31}...\left(\mathrm{i}\right)$
At x = -1 
Diff. (i) on both side 
$\mathrm{y}\text{'}\text{'}(\mathrm{x}-1)+\mathrm{y}\text{'}+\mathrm{y}\text{'}=32\times 31{\mathrm{x}}^{30}$
Substitute x = -1
$\mathrm{y}\text{'}\text{'}(-2)+2\mathrm{y}\text{'}=32\times 31$
$\mathrm{y}\text{'}-\mathrm{y}\text{'}\text{'}=16\times 31=496$