Differentiation
Differentiation [ edit ] Main articles: Derivative and Differential calculus The notion of the derivative of a function or differentiability originates from the concept of approximating a function near a given point using the "best" linear approximation. This approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point � , and the slope of the line is the derivative of the function at � . A function � : � → � is differentiable at � if the limit � ′ ( � ) = lim ℎ → 0 � ( � + ℎ ) − � ( � ) ℎ exists. This limit is known as the derivative of � at � , and the function � ′ , possibly defined on only a subset of � , is the derivative (or derivative function ) of � . If the derivative exists everywhere, the function is said to be differentiable . As a simple consequence of the definition, � is continuous at � if it is differentiable there. Differentiability is therefore a stronger regularity condi