Rate of Change of Quantities

If \(y=f(x)\), then \(\frac{dy}{dx}\) (or \(f\,'(x_0)\) at a point) is the rate of change of \(y\) with respect to \(x\). When both quantities depend on time \(t\), the chain rule links their rates:

\[\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}\]

Example. The radius of a circle grows at \(0.5\) cm/s. How fast is the area growing when \(r=4\) cm? \(A=\pi r^2\Rightarrow\frac{dA}{dt}=2\pi r\frac{dr}{dt}=2\pi(4)(0.5)=4\pi\) cm²/s.

Strategy for related rates

1. Draw a diagram; name the variables and write the given rates with signs (decreasing ⇒ negative).
2. Write an equation linking the variables (area, volume, Pythagoras, similar triangles).
3. Differentiate with respect to t.
4. Substitute the values at the required instant – only after differentiating.

Economics. If \(C(x)\) is the total cost of producing \(x\) units, the marginal cost is \(C\,'(x)\), the approximate cost of one more unit. Likewise marginal revenue is \(R\,'(x)\).