Derivative of

If , then . To see that this is true, we begin with the limit definition of the derivative:

Therefore, the derivative will exist if the limit exists. Let us sketch this difference quotient in the vicinity of :

> plot((exp(h)-1)/h,h=-1..1);

And zooming in even closer to :

> plot((exp(h)-1)/h,h=-0.001..0.001);

So, it looks like the limit surely exists and equals 1. We also can look at a table of numerical values.

> with(math3):
rightlimit((exp(h)-1)/h,h=0,10);
leftlimit((exp(h)-1)/h,h=0,10);

                   limit of  (exp(h)-1)/h  as  h -> 0  from the right

                               h                     (exp(h)-1)/h

                        -----------------------------------------

                          1.0000000000               1.7182818285

                           .1000000000               1.0517091808

                           .0100000000               1.0050167084

                           .0010000000               1.0005001667

                           .0001000000               1.0000500017

                           .0000100000               1.0000050000

                           .0000010000               1.0000005000

                           .0000001000               1.0000000500

                           .0000000100               1.0000000050

                           .0000000010               1.0000000005

                    limit of  (exp(h)-1)/h  as  h -> 0  from the left

                               h                     (exp(h)-1)/h

                        -----------------------------------------

                          -1.0000000000               .6321205588

                           -.1000000000               .9516258196

                           -.0100000000               .9950166251

                           -.0010000000               .9995001666

                           -.0001000000               .9999500017

                           -.0000100000               .9999950000

                           -.0000010000               .9999995000

                           -.0000001000               .9999999500

                           -.0000000100               .9999999950

                           -.0000000010               .9999999995

And finally, we can calculate the limit in one line in Maple:

> limit((exp(h)-1)/h,h=0);

From these explorations, we see that the value is indeed 1. From the following calculation, this means that if , then and we have discovered an important rule of differentiation:

.