Introduction
We defined the derivative as:
f'(x)=\lim_{h \to 0}\frac{f(x+h) - f(x-h)}{2h}
However, this is not the common definition. The common definition is:
f'(x)=\frac{f(x)_2 - f(x)_1}{x_2 - x_1}=\lim_{h \to 0}\frac{f(x+h) - f(x)}{(x+h)-x}=\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}
One of the two points which approaches
x
is replaced by
x
itself.
The two graphs below show the difference between the two definitions.
These two definitions are different in a critical way.
For the first, the limit's left and right side will always be equal.
\lim_{h \to 0^+}\frac{f(x+h) - f(x-h)}{2h}=\lim_{h \to 0^+}\frac{f(x+(-h)) - f(x-(-h))}{2(-h)}=\lim_{h \to 0^-}\frac{f(x+h) - f(x-h)}{2h}
For the second, the limit's left and right side are not necessarily equal.
We have used the less-common definition thus far because it more intuitively represents the tangent line. However, now that we are doing weightier proofs for more complicated functions, we will use the definition which is easier to work with and more comprehensive.
In order for a derivative to exist, the second limit must exist. Therefore, its left and right side must equal each other.
A great example of the necessity of the new definition is
f(x)=|x|
.
Using the first definition, f'(0)=0
.
Using the second definition, f'(0)=\text{undefined}
. The left side of the limit is -1
and the right side is 1
.
This is demonstrated in the following graphs.
(I graphed
f(x)=|x|+0.2
so the tangent line would be visible.)