Related rates

Related rates is a differential-calculus method for finding how quickly one quantity changes by expressing it in terms of other changing quantities and differentiating with respect to time.

Implicit differentiation: A differentiation method applied to equations in which variables are not isolated as functions of one another. Differentiating a constraint with respect to time links the quantities’ rates.
Derivative: A measure of a function’s instantaneous rate of change with respect to its input. Each unknown rate in a related-rates problem is a derivative.
Ladder problem: A calculus problem in which a ladder slides along a wall and floor while its endpoint distances change. The fixed ladder length links the wall and floor distances, allowing one endpoint’s speed to determine the other’s.
Optimization: The use of mathematical methods to find maximum or minimum values under specified conditions. Related rates asks for change at an instant, while optimization seeks extreme values.
Chain rule: A rule for differentiating a composite function by multiplying derivatives of its nested functions. It converts a changing quantity’s derivative into its rate multiplied by its dependence on another variable.
Function: A relation that assigns each input exactly one output. Changing quantities are modeled as functions of time before their rates are related.
Cone filling problem: A problem finding how a liquid’s depth or volume changes as a cone-shaped container fills or empties. Similar triangles relate the liquid’s radius to its depth before volume is differentiated.
Linear approximation: An estimate of a function near a point using the tangent line at that point. It estimates nearby values rather than deriving one changing quantity’s rate from another’s.
Differentiation with respect to time: Differentiation in which time is the independent variable and changing quantities are functions of time. Related-rates equations use time derivatives such as dx/dt and dy/dt.
Variable: A symbol representing a quantity that can take different values. Choosing variables makes each changing length, angle, or volume explicit.

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