The power rule is a fantastic "shortcut" for finding the derivatives of basic polynomials. Between the power rule and the basic definition of the derivative of a constant, a great number of polynomial derivatives can be identified with little effort  often in your head!
Constant Derivatives and the Power Rule
In this lesson, we will develop formulas and theorems that will calculate derivatives in more efficient and quick ways. Look for these theorems in boxes throughout the lesson.
The Derivative of a Constant
Theorem: If where c is a constant, then . Proof: . 

Theorem: If is a constant and is differentiable at all , then . In simpler notation 

The Power Rule
Theorem: (The Power Rule) If n is a positive integer, then for all real values of x


Examples
Example 1
Find for .
If for all , then for all .
We can also write .
Example 2
Find the derivative of .
..... Restate the function
..... Apply the commutative law
..... Apply the power Rule
..... Simplify
Example 3
Find the derivative of .
..... Restate
..... Rules of exponents
..... By the commutative law
..... Apply the power rule
..... Simplify
..... Simplify again
..... Use rules of exponents
Example 4
Find the derivative of .
Special application of the power rule:
Example 5
Find the derivative of .
Restate the function:
Using rules of exponents (from algebra):
Apply the power rule:
Simplify:
Rules of exponents:
Simplify:
Example 6
Find the derivative of .
Restate the function:
Rules of exponents:
Power rule:
Simplify:
Rules of exponents:
Review
 State the power rule.
Find the derivative:
 Given , find the derivative when .
 Given , what is ?
 when
 Given , what is ?
Review (Answers)
To see the Review answers, open this PDF file and look for section 8.9.