Answer
96
Work Step by Step
$F(x)=f(3f(4f(x)))$ _____(1)
Using chain rule differentiating (1) with respect to $x$
$F^{'}(x)=f^{'}(3f(4f(x)))\cdot [3f(4f(x))]^{'}$
Using chain rule
$F^{'}(x)=f^{'}(3f(4f(x)))\cdot [3f^{'}(4f(x))]\cdot [4f(x)]^{'}$
$F^{'}(x)=f^{'}(3f(4f(x)))\cdot [3f^{'}(4f(x))]\cdot 4f^{'}(x)$
$F^{'}(0)=f^{'}(3f(4f(0)))\cdot [3f^{'}(4f(0))]\cdot [4f^{'}(0)]$
Using given data $f(0)=0\;,\;f^{'}(0)=2$
$F^{'}(0)=f^{'}(3f(0))\cdot [3f^{'}(0)]\cdot [4 \times 2]$
$F^{'}(0)=f^{'}(0)\cdot [3\times2]\cdot [4\times 2]$
$F^{'}(0)=2\times3\times2\times4\times2$
$F^{'}(0)=96$
Hence $F^{'}(0)=96$.
