Linked Changes
If two variables are related by an equation, their rates of change over time are also related. We use Implicit Differentiation with respect to time (\(t\)).
Worked Examples
Example 1: The Balloon
Air is being pumped into a spherical balloon at a rate of \(100 \text{ cm}^3/\text{s}\). How fast is the radius increasing when the radius is \(5 \text{ cm}\)?
- Equation: \(V = \frac{4}{3}\pi r^3\).
- Differentiate with respect to \(t\): \(\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}\).
- Knowns: \(\frac{dV}{dt} = 100\), \(r = 5\).
- Solve: \(100 = 4\pi(25) \frac{dr}{dt} \implies 100 = 100\pi \frac{dr}{dt} \implies \frac{dr}{dt} = \frac{1}{\pi} \approx 0.32 \text{ cm}/\text{s}\).
- Result: \(0.32 \text{ cm}/\text{s}\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we use related rates to understand the Probability Current. If the probability of finding a particle in one region is decreasing, the probability in the neighboring region must be increasing to compensate. This is called the Continuity Equation, and it ensures that the total probability of the particle's existence is always 1. It is the quantum equivalent of fluid flow.