Instantaneous rate of change > Let $f(x) = ax + b$. > > Find the instantaneous rate of change of $f(x)$ at the following points: $1, 2, 4\text{ and }8$. Is the instantaneous rate of change for all the points $a$, because the derivative for $f'(x)= a$?

Yes, the instantaneous rate of change is $4$.

To see this (even) more explicitly, remember that the derivative $f'(x)$ is defined as $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$, so $$\frac{d}{dx} (ax+b) = \lim_{x\to 0}\frac{(a(x+h) + b) - (ax+b)}{h} = \lim_{h\to0} \frac{ah}{h} = a$$