$\begin{align}  & A=g'(1)f(1)-f'(1)g(1)=\frac{g'(1)f(1)-f'(1)g(1)}{{{\left( f(1) \right)}^{2}}}{{\left( f(1) \right)}^{2}} \\  & =\left( \frac{g(x)}{f(x)} \right)'\left| x={{1}^{{{\left( f(1) \right)}^{2}}}} \right. \\  & \frac{g(x)}{f(x)}=\frac{{{x}^{8}}16}{\left( {{x}^{2}}+2 \right)\left( {{x}^{4}}+4 \right)}={{x}^{2}}-2\Rightarrow \left( \frac{g(x)}{f(x)} \right)'=2x \\  & \Rightarrow A=2(1){{\left( f(1) \right)}^{2}}=2(1)\left( {{15}^{2}} \right)=450 \\ \end{align}$