Answer
Yes, $3g'(7)$ also exists.
Work Step by Step
Step 1. Assume we know $g(t)$ is differentiable at $t=7$ which means that $g′(7)=\lim_{x\to7}\frac{g(x)-g(7)}{x-7}$ exists.
Step 2. For a new function defined as $k(t)=3g(t)$, try to find the derivative of $k(t)$ at $x=7$ as:
$k'(7)=\lim_{x\to7}\frac{k(x)-k(7)}{x-7}=\lim_{x\to7}\frac{3g(x)-3g(7)}{x-7}=3g'(7)$
