
• When explaining the meaning of $f'(a)=b$ in context:
• If rate is NOT constant, then it need NOT be true that for any one unit change in the independent quantity (input), the dependent quantity (output) changes by $b$ units.
• If $f$ is differentiable at $a$, then there is a sufficiently small amount of change in the independent quantity (input) from $a$ where the graph of $f$ is indistinguishable from a line.
• So $f'(a)=b$ means that as the independent quantity (input) varies by a sufficiently small amount from $a$, the change in amount of the dependent quantity (output) from $f(a)$ is essentially $b$ times as much as the change in the independent quantity.