Symmetric derivative

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the symmetric derivative is an <a href="/facts/Operator_(mathematics)/1VWgEdjS">operation</a> generalizing the ordinary <a href="/facts/Derivative/7Ito70Dh">derivative</a>. 
It is defined as:

lim
          
            h
            →
            0
          
        
        
          
            
              f
              (
              x
              +
              h
              )
              −
              f
              (
              x
              −
              h
              )
            
            
              2
              h
            
          
        
        .
      
    
    {\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x-h)}{2h}}.}

The expression under the limit is sometimes called the symmetric <a href="/facts/Difference_quotient/9sC6B072">difference quotient</a>. A function is said to be symmetrically differentiable at a point x if its symmetric derivative exists at that point.
If a function is <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known counterexample is the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better <a href="/facts/Numerical_differentiation/vlbI7vhL">numerical approximation of the derivative</a> than the usual difference quotient.
The symmetric derivative at a given point equals the <a href="/facts/Arithmetic_mean/L3Pv333o">arithmetic mean</a> of the <a href="/facts/Left_and_right_derivative/UJUGB8hR">left and right derivatives</a> at that point, if the latter two both exist.: 6 
Neither <a href="/facts/Rolle%2527s_theorem/M8oMH2gp">Rolle's theorem</a> nor the <a href="/facts/Mean-value_theorem/JnjkzQb4">mean-value theorem</a> hold for the symmetric derivative; some similar but weaker statements have been proved.

Symmetric derivative open-in-new

Symmetric derivative