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Differentiation

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  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