differentiabilitás

Differentiability is a fundamental concept in calculus that describes whether a function can be "smoothed" or approximated by a straight line at a given point. A function is considered differentiable at a point if its derivative exists at that point. The derivative of a function at a point represents the instantaneous rate of change of the function at that point, or equivalently, the slope of the tangent line to the function's graph at that point.

Geometrically, a function is differentiable at a point if its graph has a well-defined, non-vertical tangent line at that point. This implies that the graph does not have any sharp corners, cusps, or breaks at that point. If a function is differentiable at a point, it is also necessarily continuous at that point. However, the converse is not always true; a function can be continuous at a point but not differentiable there, such as the absolute value function at x=0.

The existence of the derivative is formally defined using a limit. The derivative of a function f(x) at a point 'a', denoted f'(a), is given by the limit: lim (h->0) [f(a+h) - f(a)] / h, provided this limit exists. This limit represents the slope of the secant line between two points on the curve as those points become infinitesimally close to each other. Differentiability is crucial for many applications, including optimization problems, curve sketching, and the analysis of physical phenomena.