# symmetric difference quotient

## symmetric difference quotient

does not exist. It is usually a much better approximation to the derivative Q  is rational 3 The expression under the limit is sometimes called the symmetric difference quotient. The symmetric difference is commutative and associative (and consequently the leftmost set of parentheses in the previous expression were thus redundant): A B = B A , ( A B ) C = A ( B C ) . , due to discontinuity in the curve there. {\displaystyle x=0} 2 = , we have, at {\displaystyle x=0} Hence the symmetric derivative of the absolute value function exists at f ' (a) {\displaystyle f_{s}(x)} − and is equal to zero, even though its ordinary derivative does not exist at that point (due to a "sharp" turn in the curve at . 1 {\displaystyle f(x)=1/x^{2}} x But the second symmetric derivative exists for ). which is defined by. In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. f = In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). : numerical approximation of the derivative, Approximating the Derivative by the Symmetric Difference Quotient (Wolfram Demonstrations Project), https://en.wikipedia.org/w/index.php?title=Symmetric_derivative&oldid=982556563, Creative Commons Attribution-ShareAlike License. 1 for positive and negative values of ) The symmetric difference quotient is the average of the difference quotients for positive and negative values of h . a x quotient. The function difference divided by the point difference is known as "difference quotient": Δ F ( P ) Δ P = F ( P + Δ P ) − F ( P ) Δ P = ∇ F ( P + Δ P ) Δ P . s x For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient. − f'(a) . 1 is the ) As a counterexample, the symmetric derivative of f(x) = |x| has the image {−1, 0, 1}, but secants for f can have a wider range of slopes; for instance, on the interval [−1, 2], the mean value theorem would mandate that there exist a point where the (symmetric) derivative takes the value Inc. Home Contents Index Top | {\displaystyle f(x)={\begin{cases}1,&{\text{if }}x{\text{ is rational}}\\0,&{\text{if }}x{\text{ is irrational}}\end{cases}}}. {\displaystyle {\frac {|2|-|-1|}{2-(-1)}}={\frac {1}{3}}} x 0 ∈ x The sign function is not continuous at zero and therefore the second derivative for 0 {\displaystyle x=0} − If the symmetric derivative of f has the Darboux property, then the (form of the) regular mean value theorem (of Lagrange) holds, i.e. | f The symmetric difference quotient is the average of the difference quotients The symmetric difference quotient is a formula that gives an approximation of the derivative of a function, f ( x ). 0 f {\displaystyle x=0} x If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x.