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A differentiable function. In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain.In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain.
A function of a real variable is differentiable at a point of its domain, if its domain contains an open interval containing , and the limit = (+) exists. [2] This means that, for every positive real number , there exists a positive real number such that, for every such that | | < and then (+) is defined, and | (+) | <, where the vertical bars denote the absolute value.
Product rule: For two differentiable functions f and g, () = +. An operation d with these two properties is known in abstract algebra as a derivation . They imply the power rule d ( f n ) = n f n − 1 d f {\displaystyle d(f^{n})=nf^{n-1}df} In addition, various forms of the chain rule hold, in increasing level of generality: [ 12 ]
But the zero function is not a weak derivative of c, as can be seen by comparing against an appropriate test function . More theoretically, c does not have a weak derivative because its distributional derivative, namely the Cantor distribution, is a singular measure and therefore cannot be represented by a function.
If the function is differentiable, the minima and maxima can only occur at critical points or endpoints. This also has applications in graph sketching: once the local minima and maxima of a differentiable function have been found, a rough plot of the graph can be obtained from the observation that it will be either increasing or decreasing ...
A function is (totally) differentiable if its total derivative exists at every point in its domain. Conceptually, the definition of the total derivative expresses the idea that d f a {\displaystyle df_{a}} is the best linear approximation to f {\displaystyle f} at the point a {\displaystyle a} .
This function is continuous on the closed interval [−r, r] and differentiable in the open interval (−r, r), but not differentiable at the endpoints −r and r. Since f (− r ) = f ( r ) , Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero.
A function is differentiable at an interior point a of its domain if and only if it is semi-differentiable at a and the left derivative is equal to the right derivative. An example of a semi-differentiable function, which is not differentiable, is the absolute value function () = | |, at a = 0.