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For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient. [3] 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
The simplest method is to use finite difference approximations. A simple two-point estimation is to compute the slope of a nearby secant line through the points (x, f(x)) and (x + h, f(x + h)). [1] Choosing a small number h, h represents a small change in x, and it can be either positive or negative.
Difference quotients may also find relevance in applications involving Time discretization, where the width of the time step is used for the value of h. The difference quotient is sometimes also called the Newton quotient [10] [12] [13] [14] (after Isaac Newton) or Fermat's difference quotient (after Pierre de Fermat). [15]
The arithmetic difference is h = 3, as established above. Given the number of pairwise differences needed to reach the constant, it can be surmised this is a polynomial of degree 3 . Thus, using the identity above: 648 = a ⋅ 3 3 ⋅ 3 ! = a ⋅ 27 ⋅ 6 = a ⋅ 162 {\displaystyle 648=a\cdot 3^{3}\cdot 3!=a\cdot 27\cdot 6=a\cdot 162}
For arbitrary stencil points and any derivative of order < up to one less than the number of stencil points, the finite difference coefficients can be obtained by solving the linear equations [6] ( s 1 0 ⋯ s N 0 ⋮ ⋱ ⋮ s 1 N − 1 ⋯ s N N − 1 ) ( a 1 ⋮ a N ) = d !
In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative ′ of a function at a point : f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}.}
On the premiere episode of her new podcast, Khloé Kardashian recalled her "crazy, drunk" wrestling match with Scott Disick during Kim Kardashian and Kanye West's 2014 rehearsal dinner
This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The second line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a + h.