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The graph of =, with a straight line that is tangent to (,). The slope of the tangent line is equal to . (The axes of the graph do not use a 1:1 scale.) The derivative of a function is then simply the slope of this tangent line.
Specifically, draw a diagonal line connecting two points on the diagram so that every other point is either on or to the right and above it. There is at least one such line if the curve passes through the origin. Let the equation of the line be qα+pβ=r. Suppose the curve is approximated by y=Cx p/q near the origin.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
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.
The definitions are applied to graphs as follows. If a function (a -cochain) is defined at the nodes of a graph: ,,, … then its exterior derivative (or the differential) is the difference, i.e., the following function defined on the edges of the graph (-cochain):
A line, usually vertical, represents an interval of the domain of the derivative.The critical points (i.e., roots of the derivative , points such that () =) are indicated, and the intervals between the critical points have their signs indicated with arrows: an interval over which the derivative is positive has an arrow pointing in the positive direction along the line (up or right), and an ...
If it is positive then the graph has an upward concavity, and, if it is negative the graph has a downward concavity. If it is zero, then one has an inflection point or an undulation point . When the slope of the graph (that is the derivative of the function) is small, the signed curvature is well approximated by the second derivative.
A line graph has an articulation point if and only if the underlying graph has a bridge for which neither endpoint has degree one. [2] For a graph G with n vertices and m edges, the number of vertices of the line graph L(G) is m, and the number of edges of L(G) is half the sum of the squares of the degrees of the vertices in G, minus m. [6]