Diferentieeritav

Diferentieeritav is a term used in mathematics, particularly in calculus, to describe a function that possesses a derivative at a given point. A function is considered differentiable at a point if its graph has a well-defined, non-vertical tangent line at that point. This means that the rate of change of the function is precisely defined at that specific location.

For a function f(x) to be differentiable at a point 'a', the limit of the difference quotient as 'h' approaches zero must exist. This limit is represented as lim (h->0) [f(a+h) - f(a)] / h. If this limit exists and is a finite number, then the function is differentiable at 'a', and the value of the limit is the derivative of the function at that point, denoted as f'(a).

Differentiability implies continuity. A function that is differentiable at a point is also continuous at that point. However, the converse is not always true; a function can be continuous at a point but not differentiable there. Common examples of points where a continuous function might not be differentiable include sharp corners, cusps, or vertical tangents. The concept of differentiability is fundamental to many areas of mathematics and its applications, enabling the study of rates of change, optimization, and the behavior of functions.