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The formula simplifies: = ′ (). The unit tangent vector determines the orientation of the curve, or the forward direction, corresponding to the increasing values of the parameter. The unit tangent vector taken as a curve traces the spherical image of the original curve.
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.
In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).
In physics this theorem is one of the ways of defining a conservative force. By placing φ as potential, ∇ φ is a conservative field . Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers . So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when
As shown below, the second-derivative test is mathematically identical to the special case of n = 1 in the higher-order derivative test. Let f be a real-valued, sufficiently differentiable function on an interval I ⊂ R {\displaystyle I\subset \mathbb {R} } , let c ∈ I {\displaystyle c\in I} , and let n ≥ 1 {\displaystyle n\geq 1} be a ...
Then the same equation = ′ + holds with the same definition of Δy, but instead of ε being infinitesimal, we have = (treating x and f as given so that ε is a function of Δx alone). See also [ edit ]