Estimating a Derivative (numerically)

We can estimate f '(a) numerically using three difference quotients which are defined as follows:

1) Forward Difference Quotient

2) Backward Difference Quotient

3) Symmetric Defference Quotient

And we can also recognize that symmetric difference quotient is equal to
 

which is just the average of  forward and backward difference quotients.

Many calculators use the symmetric difference quotient in finding derivatives.

The exact value of f '(1) is 4.

In fact, f (x)=|x| does not have a derivative at x=0 because the function has different left-hand and right-hand limits at x=0.

                                                     There is a corner at x=0

Sometimes a wrong result is obtained using the symmetric difference quotient.
So, we shoud check differentiability before using symmetric difference quotient.