Answer
To find the rate of change of function $g(t)$ at $t=t_0$, the following limit must be calculated: $$\lim_{h\to0}\frac{\Delta y}{\Delta x}=\lim_{h\to0}\frac{g(t_0+h)-g(t_0)}{h}$$
Work Step by Step
To find the rate of change of the function $y=g(t)$ at $t=t_0$, a limit of average rate of change must be calculated.
Recall the average rate of change formula for $y=g(t)$ over the interval from $t=t_0$ to $t=t_1$ and $t_1-t_0=h$:
$$\frac{\Delta y}{\Delta x}=\frac{g(t_1)-g(t_0)}{t_1-t_0}=\frac{g(t_0+h)-g(t_0)}{h}$$
To calculate the limit of average rate of change for $g(t)$ at $t=t_0$, we calculate the limit of the function of the rate of change above as $h\to0$:
$$\lim_{h\to0}\frac{\Delta y}{\Delta x}=\lim_{h\to0}\frac{g(t_0+h)-g(t_0)}{h}$$
