differentiëble

Differentiëble is a term used in calculus to describe a function that can be differentiated. A function is considered differentiable if its derivative exists at a given point or over a given interval. The derivative of a function at a point represents the instantaneous rate of change of the function at that point, which geometrically corresponds to the slope of the tangent line to the function's graph at that point.

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

A function that is differentiable at a point is necessarily continuous at that point. However, the converse is not always true; a function can be continuous at a point but not differentiable. A common example of this is the absolute value function at x=0. While it is continuous at x=0, it has a sharp corner, meaning the slope from the left differs from the slope from the right, and thus the derivative does not exist. Differentiability is a crucial concept for many areas of mathematics, physics, and engineering, enabling the analysis of how quantities change.