Rate of Change of Quantities

6.2 Rate of Change of Quantities

The Fundamental Idea

When one quantity $y$ depends on another quantity $x$ through $y = f(x)$, the derivative $\frac{dy}{dx}$ (or $f'(x)$) gives the instantaneous rate of change of $y$ with respect to $x$ — how fast $y$ is changing at a particular value of $x$. Evaluated at $x = x_0$:

The Chain Rule Connection

Often both $x$ and $y$ vary with respect to a third variable, usually time $t$: $x = f(t)$ and $y = g(t)$. We cannot write $y$ directly as a function of $x$, but the Chain Rule still gives:

Sign of the Rate of Change

• Positive $\frac{dy}{dx}$: $y$ increases as $x$ increases.
• Negative $\frac{dy}{dx}$: $y$ decreases as $x$ increases.

Economic Rates

A Quick Worked Illustration

Suppose a circular ripple spreads outward so its radius $r$ grows at $\dfrac{dr}{dt} = 3\text{ cm/s}$. How fast is the enclosed area $A = \pi r^2$ growing when $r = 4\text{ cm}$?

Differentiate $A$ with respect to $t$ using the Chain Rule (since $A$ depends on $r$, and $r$ depends on $t$): …