Related Rates
Related rates problems use implicit differentiation with respect to time to find how the rate of change of one quantity relates to the rate of change of another quantity, given that both are functions of time. A geometric or physical relationship between the quantities is differentiated with respect to time, and known rates are substituted to find the unknown rate. These problems are extensively used in physics and engineering, for instance to relate the rate at which a ladder slides down a wall to the rate at which the base moves away from the wall.
Key Formula
dV/dt = (dV/dr) · (dr/dt)
LaTeX: \frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}
| Symbol | Meaning | Unit |
|---|---|---|
| dV/dt | rate of change of volume with respect to time | m³/s |
| dV/dr | rate of change of volume with respect to radius | m² |
| dr/dt | rate of change of radius with respect to time | m/s |
| t | time | s |
Worked Example
Problem
A spherical balloon is being inflated so that its radius increases at 2 cm/s. How fast is the volume increasing when the radius is 5 cm? (V = (4/3)πr³)
Solution
Step 1: Write the volume formula: V = (4/3)πr³. Step 2: Differentiate both sides with respect to time t: dV/dt = (4/3)π · 3r² · (dr/dt) = 4πr² · (dr/dt). Step 3: Substitute r = 5 cm and dr/dt = 2 cm/s: dV/dt = 4π(5)²(2) = 4π · 25 · 2 = 200π.
Answer
dV/dt = 200π ≈ 628.3 cm³/s
Common Related Rates Scenarios and Key Relationships
| Scenario | Key Equation | Variables | Differentiate With Respect To |
|---|---|---|---|
| Expanding sphere | V = (4/3)πr³ | V, r, t | Time t |
| Expanding circle | A = πr² | A, r, t | Time t |
| Sliding ladder | x² + y² = L² | x, y, t | Time t |
| Conical water tank | V = (1/3)πr²h | V, r, h, t | Time t |
| Shadow length | Similar triangles | object height, shadow, position, t | Time t |
Interactive Tools
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Related Terms
Implicit Differentiation
Implicit differentiation is a technique for finding the derivative dy/dx when a relationship between x and y is defined implicitly by an equation, rather than expressed as y = f(x) explicitly. It involves differentiating both sides of the equation with respect to x and applying the chain rule whenever y is differentiated, since y is a function of x. This method is essential for finding slopes of curves defined by circles, ellipses, and other relations that cannot be easily solved for y.
Chain Rule
The chain rule is a differentiation rule used to compute the derivative of a composite function, stating that the derivative of f(g(x)) equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. It is one of the most widely applied rules in calculus, essential whenever a function is "nested" inside another. The chain rule is critical in physics for related rates problems, in machine learning for backpropagation, and in multivariable calculus for total derivatives.
Derivative (calculus)
The derivative of a function at a point measures the instantaneous rate of change of the function's output with respect to its input at that point, and geometrically represents the slope of the tangent line to the function's graph. Derivatives are defined as the limit of the difference quotient as the interval shrinks to zero. They are central to physics, engineering, economics, and all sciences wherever rates of change or optimisation are relevant.
The term "related rates" is a descriptive English phrase indicating that two rates of change are related through a common equation. The technique is an application of implicit differentiation, which dates to the 17th century. The specific pedagogical category "related rates" was formalised in calculus textbooks during the 19th and 20th centuries.