Concept preview

What you will be able to do

1. 01Interpret a derivative $\dfrac{dy}{dx}$ or $f'(x)$ as an instantaneous rate of change in a NON-motion context — population growth, economics, biology, and fluid flow — and attach the correct units of (output units) per (input unit).
2. 02Build a rate model from a verbal description: identify the quantity that is changing, the variable it changes with respect to, and translate "how fast is it changing" into a derivative.
3. 03Read the sign and magnitude of an applied rate to say whether a quantity is increasing or decreasing and how quickly, and distinguish an average rate over an interval from the instantaneous rate at a point.

Suppose a quantity $Q$ depends on an input variable — call it $t$ for time, or $x$ for a count of items. The derivative $\dfrac{dQ}{dt}$ (or $Q'(t)$) is the instantaneous rate of change of $Q$: it answers "at this exact moment, how fast is $Q$ changing per unit of the input?" This is the same limit of average rates you built in Unit 2, $\displaystyle\lim_{h\to 0}\dfrac{Q(t+h)-Q(t)}{h}$, but now $Q$ might be a number of bacteria, a cost in dollars, a concentration in moles per litre, or a volume in litres. The idea travels; only the labels differ.