DIFFERENTIABLE FUNCTION Meaning and
Definition
-
A differentiable function is a mathematical concept used in calculus to describe a function that possesses defined and continuous derivatives across its entire domain. More specifically, a function f(x) defined on an open interval I is said to be differentiable at a point c in I if the derivative of f(x) exists at c. This means that the slope of the tangent line to the graph of the function at that point is well-defined.
Differentiability implies the continuity of the function at the given point, as the function must be continuous in order to have a derivative. Therefore, a differentiable function is a subset of continuous functions.
The derivative of a differentiable function is a new function that represents the rate of change of the original function at each point. It is often denoted as f'(x) or dy/dx, symbolizing the instantaneous rate of change of y with respect to x.
Functions that are differentiable in all points of their respective domains are called differentiable functions. These functions exhibit smooth and continuous behavior, allowing for the application of various calculus techniques, such as finding critical points, local extrema, and solving optimization problems. This concept of differentiability is a crucial foundation for many advanced mathematical concepts and calculations in areas such as physics, engineering, economics, and more.
Etymology of DIFFERENTIABLE FUNCTION
The term "differentiable" originated from the Latin word "differentiabilis", which combines "differentia" (meaning "difference" or "distinction") and the suffix "-abilis" (indicating capability or capacity). In mathematics, the concept of differentiability refers to the property of a function that allows for the calculation of its derivative at every point within its domain. Thus, a differentiable function is one that can be distinguished or differentiated, with its derivative providing information about its rate of change at each point.
