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It is defined as: A function is symmetrically differentiable at a point x if its symmetric derivative exists at that point. Let the symmetric derivative of f at x be, lim h->0 (f(x+h) + f(x-h) - 2f(x))/(h) Assume f is continuous on the interval [0,1] and the symmetric derivative exists at all points in (0,1). Received by the editors July 26, 1988. In mathematics, the symmetric derivative is an operation related to the ordinary derivative.. Symmetric matrix is used in many applications because of its properties. When fs(x) = fs(x), whether finite or infinite, the common value is denoted by fs(x) and is called the symmetric derivative of f at x. This theorem seems to run contrary to the following example. The new function obtained by differentiating the derivative is called the second derivative. The eigenvalue of the symmetric matrix should be a real number. Monotonicity, symmetric derivative, scattered sets, symmetric covers, symmetric continuity, symmetric derivation bases. and the lower symmetric derívate fs(x) is the corresponding limit inferior. The derivative of velocity is the rate of change of velocity, which is acceleration. 1980 Mathematics Subject Classification (1985 Revision). Prove that there exists a point x in the open interval (0,1) where the ordinary derivative exists. For example, suppose A = µ 1 2 1 3 ¶: We ﬂrst calculate the characteristic polynomial, det(A¡‚I) = det µ 1¡‚ 2 1 3¡‚ ¶ = (1 ¡‚)(3¡‚)¡2 = ‚2 ¡4‚+1: Primary 26A24; Secondary 26A48. For example for vectors, each point in has a basis , so a vector (field) has components with respect to this basis: Covariant differentiation¶ The derivative of the basis vector is a vector, thus it can be written as a linear combination of the basis vectors: ... At the beginning we used the usual trick that is symmetric but is antisymmetric. Key words and phrases. 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