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The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
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.
When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once.
The graph of the Dirac delta is usually thought of as following the whole x-axis ... with x = (x 1, x 2, ..., x n), one ... The derivative of the Dirac delta ...
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.
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
Therefore, the true derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: ...
If x 0 is an interior point in the domain of a function f, then f is said to be differentiable at x 0 if the derivative ′ exists. In other words, the graph of f has a non-vertical tangent line at the point (x 0, f(x 0)). f is said to be differentiable on U if it is differentiable at every point of U.
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