Search results
Results From The WOW.Com Content Network
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.
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 ]
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.
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.
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 ...
The simplest setting in which strict differentiability can be considered, is that of a real-valued function defined on an interval I of the real line. The function f:I → R is said strictly differentiable in a point a ∈ I if
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. . According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite direc