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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.
Graph of the function 3x 3-5x 2 +8 (black) and its first (9x 2-10x, red) and second (18x-10, blue) derivatives.An x value where the y value of the red, or the blue, curve vanishes (becomes 0) gives rise to a local extremum (marked "HP", "TP"), or an inflection point ("WP"), of the black curve, respectively.
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if is a holomorphic function, real-valued on the real line, which can be evaluated at points in the complex plane near , then there are stable methods.
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 ...
For this reason, the derivative is sometimes called the slope of the function f. [50]: 61–63 Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x 2 be the squaring function. The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is ...
In mathematics, calculus on finite weighted graphs is a discrete calculus for functions whose domain is the vertex set of a graph with a finite number of vertices and weights associated to the edges.
In electrostatics, φ(x) is interpreted as the electric potential, ρ(x) as electric charge density, and the normal derivative (′) ^ ′ as the normal component of the electric field. If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that G ( x , x ′) vanishes when either x or x′ is ...
Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.