Diffrenciálható

Diffrenciálható is a Hungarian term that translates to "differentiable" in English. In mathematics, a function is said to be differentiable at a point if its derivative exists at that point. This means that the function has a well-defined rate of change at that point, and can be approximated by a linear function (its tangent line) near that point.

A function is differentiable over an interval if it is differentiable at every point within that interval. Differentiability is a crucial concept in calculus, as it allows for the use of differentiation techniques to analyze the behavior of functions. For example, differentiable functions are continuous, and their derivatives can be used to find local maxima and minima, as well as to analyze the concavity of the function.

Not all functions are differentiable. For instance, functions with sharp corners or vertical tangents, such as the absolute value function or the function f(x) = |x|, are not differentiable at x = 0. Similarly, functions with discontinuities, such as the step function, are not differentiable at the point of discontinuity.

The concept of differentiability can be extended to higher dimensions, where it is used to study the differentiability of vector-valued functions and the differentiability of functions of several variables. In these cases, the derivative is replaced by the Jacobian matrix or the gradient, respectively.