Power Rule
The Power Rule, one of the most commonly used rules in Calculus, says:
The derivative of x^{n} is nx^{(n1)}
Example: What is the derivative of x^{2} ?
For x^{2} we use the Power Rule with n=2:
The derivative of x^{2}  =  2x^{(21)} 
=  2x^{1} 

=  2x 
Answer: the derivative of x^{2} is 2x
"The derivative of" can be shown with the little mark ’
So we get this definition:
f’(x^{n}) = nx^{(n1)}
Example: What is the derivative of x^{3} ?
f’(x^{3}) = 3x^{3−1} = 3x^{2}
"The derivative of" can also be written \frac{d}{dx}
Example: What is (1/x) ?
1/x is also x^{1}
Using the Power Rule with n = −1:
x^{n} = nx^{n−1}
x^{−1} = −1x^{−1−1} = −x^{−2}
How to Remember
"multiply by power
then reduce power by 1"
A Short Table
Here is the Power Rule with some sample values. See the pattern?
f  f’(x^{n}) = nx^{(n1)}  f’ 

x  1x^{(11)} = x^{0}  1 
x^{2}  2x^{(21)} = 2x^{1}  2x 
x^{3}  3x^{(31)} = 3x^{2}  3x^{2} 
x^{4}  4x^{(41)} = 4x^{3}  4x^{3} 
etc...  
And for negative exponents:  
x^{1}  1x^{(11)} = x^{2}  x^{2} 
x^{2}  2x^{(21)} = 2x^{3}  2x^{3} 
x^{3}  3x^{(31)} = 3x^{4}  3x^{4} 
etc... 