Variations on Leibniz method for finding derivatives.?

Question

We know from basic differential calculus that:
lim_{h → 0} [ f(x + h) - f(x) ] / h

But I have also seen the variations:
lim_{h → 0} [ f(x) - f(x - h) ] / h
lim_{h → 0} [ f(x + h) - f(x - h) ] / 2h

Do these variations hold any practicality? Why might one ever prefer one method over the other?

If I am ever forced to use this limit evaluation to determine a derivative, I am sure the first, primary one would suffice.

Answer 1

Actually, there is a practical use for the third one. While in calculus you will usually be given nice symbolic formulas where the derivative can be computed exactly, in many real-world situations, you won't have exact formulas available and will have to estimate the derivative by taking an actual difference quotient (e.g. to recover the velocity of a vehicle from a graph of position vs. time). Now, if the function has a large second derivative near the point x, then the usual difference quotient will be consistently larger or smaller than the actual derivative (larger if the second derivative and h have the same sign, smaller if they have opposite signs). This is intuitively because actual evaluation of the difference quotient for any particular h gives you the average slope over the region between x and x+h, and if the slope is increasing rapidly, that will always be larger than the slope at x if h is positive, and always smaller if h is negative (vise versa if the slope is decreasing). The third formula, on the other hand, has the advantage that by taking the average slope over a region that includes both steeper and shallower portions of the function, these effects tend to cancel each other out, leaving you with a difference quotient closer to the actual derivative. Consider the following simple example:

Numerical evaluation of d(x³)/dx at x=5:
h | (f(x+h) - f(x))/h | error | (f(x+h) - f(x-h))/h | error
1 .......... 91 ............ 16 ............ 76 .............. 1
.1 ....... 76.51 ........ 1.51 ....... 75.01 ......... 0.01
.01 .. 75.1501 ..... 0.1501 ... 75.0001 ...... 0.0001

As you can see, the third formula converges to the actual value of the derivative much more quickly than the definitional formula. As such, when estimating the derivative using an actual difference quotient, the third formula is actually preferable.