Answer
$6x(\ln 4)4^{x^{2}+2}$
Work Step by Step
$y=3\cdot 4^{x^{2}+2}$
$y=3g(x),\qquad g(x)=4^{x^{2}+2}$,
$\displaystyle \frac{dy}{dx}=3\cdot\frac{d}{dx}[g(x)],$
g(x) is a composite function,
$g(x)=4^{h(x)},\qquad h(x)=x^{2}+2, \quad h^{\prime}(x)=2x$
Using$\quad \color{blue}{ \displaystyle \frac{d}{dx}(a^{h(x)})=(\ln a)a^{h(x)}\cdot h^{\prime}(x)}$,
$\displaystyle \frac{dy}{dx}=3\cdot\ln 4\cdot 4^{x^{2}+2}\cdot 2x$
$\ \ \ \ \ \ =6x\cdot\ln 4\cdot 4^{x^{2}+2}$
