Answer
$f_x=\frac{3}{2\sqrt{3x+4t}}$, $f_t=\frac{2}{\sqrt{3x+4t}}$.
Work Step by Step
$f(x,t)=\sqrt{3x+4t}$
To find $f_x$ we treat $t$ as a constant and differentiate with respect to $x$.
$f_x=\frac{3}{2\sqrt{3x+4t}}$
To find $f_t$ we treat $x$ as a constant and differentiate with respect to $t$.
$f_t=\frac{2}{\sqrt{3x+4t}}$