In Section 2.2 we defined the derivative at $x=a\text{,}$ $f'(a)\text{,}$ of an abstract function $f(x)\text{,}$ to be its instantaneous rate of change at $x=a\text{:}$
• If you are moving along a line and $x(t)$ is your position on the line at time $t\text{,}$ then your rate of change of position, $x'(t)\text{,}$ is your velocity. If, instead, $v(t)$ is your velocity at time $t\text{,}$ then your rate of change of velocity, $v'(t)\text{,}$ is your acceleration. We shall explore this further in Section 3.1.
• If $P(t)$ is the size of some population (say the number of humans on the earth) at time $t\text{,}$ then $P'(t)$ is the rate at which the size of that population is changing. It is called the net birth rate. We shall explore it further in Section 3.3.3.
• A capacitor is an electrical component that is used to repeatedly store and release electrical charge (say electrons) in an electronic circuit. If $Q(t)$ is the charge on a capacitor at time $t\text{,}$ then $Q'(t)$ is the instantaneous rate at which charge is flowing into the capacitor. That's called the current. The standard unit of charge is the coulomb. One coulomb is the magnitude of the charge of approximately $6.241 \times 10^{18}$ electrons. The standard unit for current is the amp. One amp represents one coulomb per second.