AN EXAMPLE OF THE LIMIT METHOD
FOR FINDING THE DERIVATIVE OF f
Please review the general discussion before trying this example.
We pick a relatively simple example here : f(x) = x^{3} .
First recall the definition of "derivative" for any function f :
f ' (x) = 

h0 

f(x+h)  f(x)
(x+h)  x 

Now we interpret the functional notation for our particular function f(x) = x^{3} ;
at this point, you may want to review the binomial theorem :
f ' (x) = 

h0 

(x+h)^{3}  x^{3} 

(x+h)  x 

Now we must rewrite the fraction, since both top and bottom approach zero :
= 

h0 

[x^{3} + 3x^{2}h + 3xh^{2} + h^{3}]  x^{3} 

(x+h)  x 

Now we cancel two pairs of numbers, colored red above, and factor the top :
= 

h0 

3x^{2}h + 3xh^{2} + h^{3} 

h 

= 

h0 

h[3x^{2} + 3xh + h^{2}] 

h 

At this point, both top and bottom are ZERO if h = 0.
However, if h is NOT ZERO, we cancel the two red h above:
= 

h0 

[3x^{2} + 3xh + h^{2}] 
And now the limit can be readily seen: the last two terms shrink to ZERO while the first term remains fixed.
= 3x^{2} + 0 + 0 = 3x^{2} .
Thus, f ' (x) = 3x^{2}



This page last updated 13 February 2016 
