Rate of Change of Quantities

Whenever a quantity $y$ is expressed as a function of another quantity $x$, i.e. $y = f(x)$, the derivative $\dfrac{dy}{dx}$ (evaluated at a particular $x = x_0$) measures the instantaneous rate of change of $y$ with respect to $x$ at that point -- how fast $y$ is changing per unit change in $x$, at the instant $x = x_0$.

Rates with respect to time. In most physical and real-life situations, both $x$ and $y$ are themselves functions of a third variable, time $t$: $x = x(t)$, $y = y(t)$, and the two are related by some equation, e.g. $y = f(x)$. Differentiating such a relation with respect to $t$ using the chain rule gives

which lets the rate of change of $y$ be found from the rate of change of $x$ (and vice-versa), without ever needing $x$ and $y$ as explicit functions of $t$ separately. This technique -- differentiating a geometric relation with respect to time, then substituting known numerical rates -- is called related rates, and it is the single method underlying every problem in this section.

Worked method (illustrative). Suppose a variable $x$ (e.g. the side of a square) changes with time so that at some instant its rate of change $\dfrac{dx}{dt}$ is known, and a second quantity $y$ (e.g. the square's area, $y = x^2$) depends on $x$. Then

Substituting the known value of $x$ at that instant, and the known value of $\dfrac{dx}{dt}$, gives the numerical rate at which $y$ is changing at that instant (see Example 1).

Sign convention. A positive rate ($dx/dt > 0$) means the quantity is increasing with time; a negative rate ($dx/dt < 0$) means it is decreasing. When a problem states that a ladder's foot is being "pulled away," or a balloon is being "inflated," the corresponding rate is positive; when it asks how fast something is "decreasing," the rate found should come out negative, confirming the direction of change (see Exercise: Rate of Change of Quantities, Q2, the sliding-ladder problem, where $dy/dt$ works out negative because the height on the wall is falling).

Marginal cost and marginal revenue (an economic application of the same idea): if $C(x)$ is the total cost of producing $x$ units, the marginal cost is $\dfrac{dC}{dx}$, the approximate cost of producing one additional unit at that production level; similarly the marginal revenue is $\dfrac{dR}{dx}$ for total revenue $R(x)$. Both are simply the rate of change of cost (or revenue) with respect to the number of units -- the same derivative-as-a-rate idea applied to a quantity that is really discrete but is treated as continuous for the purpose of this approximation.