8.9: Constant Derivatives and the Power Rule
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!
Watch This
Embedded Video:
 Khan Academy: Calculus: Derivatives 3
Guidance
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


Example A
Find for
Solution
 If for all , then for all
 We can also write
Example B
Find the derivative of
Solution
 ..... Restate the function
 ..... Apply the Commutative Law
 ..... Apply the Power Rule
 ..... Simplify
Example C
Find the derivative of
Solution
 ..... Restate
 ..... Rules of exponents
 ..... By the Commutative law
 ..... Apply the Power Rule
 ..... Simplify
 ..... Simplify again
 ..... Use rules of exponents
Vocabulary
A theorem is a statement accepted to be true based on a series of reasoned statements already accepted to be true. In the context of this lesson, a theorem is a rule that allows a quick calculation of the derivative of functions of different types.
A proof is a series of true statements leading to the acceptance of truth of a more complex statement.
Guided Practice
Questions
Find the derivatives of:
1)
2)
3)
4)
Solutions
1) By the power rule:
 If then
2) Special application of the power rule:
3) Restate the function:
 Using rules of exponents (from Algebra):
 Apply the Power Rule:
 Simplify:
 Rules of exponents:
 Simplify:
4) Restate the function:
 Rules of exponents:
 Power Rule:
 Simplify:
 Rules of exponents:
Practice
 State the Power Rule.
Find the derivative:
 given when
 given what is
 when
 given what is
derivative
The derivative of a function is the slope of the line tangent to the function at a given point on the graph. Notations for derivative include , , , and \frac{df(x)}{dx}.proof
A proof is a series of true statements leading to the acceptance of truth of a more complex statement.theorem
A theorem is a statement that can be proven true using postulates, definitions, and other theorems that have already been proven.Image Attributions
Description
Learning Objectives
Here you will learn how to quickly identify the derivatives of constant terms, and you will explore the use of the power rule for finding the derivatives of higherorder functions.
Difficulty Level:
At GradeSubjects:
Concept Nodes:
Date Created:
Nov 01, 2012Last Modified:
Jun 08, 2015Vocabulary
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