Understanding Differentiability in Mathematics | The Concept, Definition, and Significance for Functions

What is the definition of “differentiable”?

In mathematics, a function is considered differentiable if it has a derivative at every point in its domain

In mathematics, a function is considered differentiable if it has a derivative at every point in its domain. The derivative represents the rate of change of the function at any given point and provides information about the function’s behavior.

Formally, let f(x) be a function defined on an interval I. The function f(x) is differentiable at a point c in I if the derivative of f(x) exists at c. This means that the limit of the difference quotient, as x approaches c, exists:

f'(c) = lim (h -> 0) [(f(c + h) – f(c))/h]

If this limit exists, it gives the instantaneous rate of change of the function at point c. Geometrically, it represents the slope of the tangent line to the graph of the function at that point.

If a function is differentiable on its entire domain, it is referred to as a differentiable function. However, note that not all functions are differentiable at every point. For example, functions with sharp corners, cusps, or vertical tangent lines are not differentiable at those points.

Differentiation is an important tool in calculus and is used to analyze rates of change, find critical points, determine concavity, and solve optimization problems, among other applications.

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