Derivative as a rate of change

Rate of change

average: $\frac{f (ω + ℎ) - f (ω)}{ℎ}$
instantaneous: $f^{'} (ω) = \lim_{ℎ \to 0} \frac{f (ω - ℎ) - f (ω)}{ℎ}$

Physics

$p (t)$ position in function of time

velocity: $v (t) = p^{'} (t)$
speed: $s (t) = | v (t) | = | p^{'} (t) |$
acceleration: $a (t) = v^{'} (t) = p^{″} (t)$
jerk: $j (t) = a^{'} (t) = v^{″} (t) = p^{‴} (t)$

Economics

$c (x)$ cost in function of unit goods

average: $\frac{c (ω + ℎ) - c (ω)}{ℎ}$
marginal: $c^{'} (ω) = \lim_{ℎ \to 0} \frac{c (ω - ℎ) - c (ω)}{ℎ}$

Biology

$P (t)$ in number of entities in a population in time $t$ ; population growth

P^{'} (t)

Examples

A particle position follows $p (t) = t^{3} - 9 t^{2} + 24 t + 4$ for $0 \leq t$

What is the velocity of the particle?
When is the particle at rest?
When is the particle moving to the left and when to the right?

Suppose that it costs

c (x) = x^{3} - 6 x^{2} + 15 x

dollars to produce $x$ radiators when eight to thirty radiators are produced and that the revenue function for selling $x$ radiators is

r (x) = x^{3} - 3 x^{2} + 12 x

If you produce ten radiators a day about how extra will it cost to produce one more radiator a day? What is the estimated increase in revenue in that case?

The population of a city is tripling every five years. If its current population is ten thousand what will be its population approximately in two years?