The Power Rule The Power Rule
Introduction
The power rule allows us to find derivatives of functions of the form
f(x)=ax^n
, where
a
and
n
are constants.
For example,
f(x)=3x^2
or
f(x)=5x^3
.
Before we prove it, note the following:
(x+h)^n = x^n + \binom{n}{1} x^{n-1} h + \binom{n}{2} x^{n-2} h^2 + \binom{n}{3} x^{n-3} h^3 + \cdots + h^n
This is the binomial theorem. We will use it to prove the power rule. In that notation there is a shorthand for the binomial coefficient,
\binom{a}{b}
.
\binom{a}{b} = \frac{a!}{b!(a-b)!}
We do not need to know the binomial coefficent, but please note the following:
\binom{n}{1} = \frac{n!}{1!(n-1)!}=n
Proof
f(x)=x^n
Define the derivative.
f'(x)=\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}
Substitute
f(x)
with
x^n
.
f'(x)=\lim_{h \to 0}\frac{(x+h)^n - x^n}{h}
Expand the binomial.
f'(x)=\lim_{h \to 0}\frac{(x^n + \binom{n}{1} x^{n-1} h + \binom{n}{2} x^{n-2} h^2 + \binom{n}{3} x^{n-3} h^3 + \cdots + h^n) - x^n}{h}
Remove
x^n
and
-x^n
f'(x)=\lim_{h \to 0}\frac{\binom{n}{1} x^{n-1} h + \binom{n}{2} x^{n-2} h^2 + \binom{n}{3} x^{n-3} h^3 + \cdots + h^n}{h}
Extract
h
from the numerator.
f'(x)=\lim_{h \to 0}\frac{h(\binom{n}{1} x^{n-1} + \binom{n}{2} x^{n-2} h + \binom{n}{3} x^{n-3} h^2 + \cdots + h^{n-1})}{h}
Cancel
h
from the numerator and denominator.
f'(x)=\lim_{h \to 0}\binom{n}{1} x^{n-1} + \binom{n}{2} x^{n-2} h + \binom{n}{3} x^{n-3} h^2 + \cdots + h^{n-1}
As
h
approaches
0
, all terms with
h
will equal
0
.
f'(x)=\lim_{h \to 0}\binom{n}{1} x^{n-1} + 0 + 0 + \cdots + 0
Solve the limit.
f'(x)=\binom{n}{1} x^{n-1}
Simplify, using
\binom{n}{1}=n
.
f'(x)=n x^{n-1}
Graphs
Here is the definition of the power rule:
f(x)=x^n
f'(x)=nx^{n-1}
f(x) = x^2
f'(x)=2x
f(x) = x^3
f'(x)=3x^{2}
f(x) = x^3 - 2x^2
f'(x)=3x^2 - 4x