In calculus, the power rule is the following rule of differentiation.
Power Rule: For any real number ,
Using the rules of differentiation and the power rule, we can calculate the derivative of polynomials as follows:
Given a polynomial
the derivative of the polynomial is
Proof: Using the addition and multiplication by a constant rules for differentiation, we have
where the last line follows from the power rule. This proves the theorem.
What is the derivative of
Applying the power rule with , we have
Since , we apply the power rule with to obtain
Given the polynomial
We use the power rule to calculate the derivative of polynomial
Proof of Power Rule 2:
Recall the formal definition of a derivative
and the binomial theorem .
Then when our ,
At this point, the terms with disappear, and we are left with