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Limit definition of the derivative

Definition

The limit definition of the derivative is a mathematical expression used to find the instantaneous rate of change (slope) of a curve at a specific point. It involves taking the limit as the change in x approaches zero.

Analogy

Imagine driving along a curvy road and trying to determine your speed at an exact moment. The limit definition of the derivative is like zooming in closer and closer to that moment until you reach an infinitesimally small time interval (change in x), allowing you to calculate your instantaneous speed (derivative).

Related terms

Derivative: The derivative measures how quickly a quantity changes with respect to another quantity. It gives us information about rates of change and slopes.

Tangent line: A tangent line is a straight line that touches a curve at only one point. It represents the instantaneous rate of change (slope) of the curve at that point.

Differentiation: Differentiation is the process of finding derivatives. It involves applying rules and formulas to determine how a function changes with respect to its input variable.