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  2. Second derivative - Wikipedia

    en.wikipedia.org/wiki/Second_derivative

    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.

  3. Second partial derivative test - Wikipedia

    en.wikipedia.org/wiki/Second_partial_derivative_test

    Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at (,) since = <. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point.

  4. Derivative test - Wikipedia

    en.wikipedia.org/wiki/Derivative_test

    After establishing the critical points of a function, the second-derivative test uses the value of the second derivative at those points to determine whether such points are a local maximum or a local minimum. [1] If the function f is twice-differentiable at a critical point x (i.e. a point where f ′ (x) = 0), then:

  5. Numerical differentiation - Wikipedia

    en.wikipedia.org/wiki/Numerical_differentiation

    Given below is the five-point method for the first derivative (five-point stencil in one dimension): [10] ′ = (+) + (+) + + (), where [, +]. For other stencil configurations and derivative orders, the Finite Difference Coefficients Calculator is a tool that can be used to generate derivative approximation methods for any stencil with any ...

  6. Hermite interpolation - Wikipedia

    en.wikipedia.org/wiki/Hermite_interpolation

    The quintic Hermite interpolation based on the function (), its first (′) and second derivatives (″) at two different points (and ) can be used for example to interpolate the position of an object based on its position, velocity and acceleration.

  7. Differential calculus - Wikipedia

    en.wikipedia.org/wiki/Differential_calculus

    The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum.

  8. Symmetric derivative - Wikipedia

    en.wikipedia.org/wiki/Symmetric_derivative

    The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. [1] [2]: 6 Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.

  9. Notation for differentiation - Wikipedia

    en.wikipedia.org/wiki/Notation_for_differentiation

    for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are: [6]