dérivable

In mathematics, a function is said to be derivable if its derivative exists at a given point. The derivative of a function at a point represents the instantaneous rate of change of the function's value with respect to its input variable at that point. Geometrically, the derivative corresponds to the slope of the tangent line to the function's graph at that point.

A function f(x) is derivable at a point 'a' if the limit of the difference quotient [f(x) - f(a)] / (x - a) as x approaches 'a' exists and is finite. This limit is denoted as f'(a). For a function to be derivable at a point, it must also be continuous at that point. However, continuity does not guarantee derivability. For instance, the absolute value function |x| is continuous at x=0 but not derivable there due to a sharp corner.

The concept of derivability is fundamental in calculus and has wide-ranging applications in physics, engineering, economics, and many other fields where rates of change are important. Functions that are derivable over an interval are often referred to as differentiable functions. If a function is derivable at every point in an interval, it is said to be differentiable on that interval. Higher-order derivatives, such as the second derivative, can also be computed for functions that are repeatedly derivable.