Power Rule
Power Rule
In calculus, thepower ruleis the following rule of differentiation.
Power Rule:For any real number ,
Using therules of differentiationand 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.
Worked Examples
What is the derivative of
Applying the power rule with , we have
What is
Since , we apply the power rule with to obtain
What is
We have
Given the polynomial
what is
We use the power rule to calculate the derivative of polynomial
Proof of Power Rule 1
Proof of Power Rule 1:
Using the identity we differentiate both sides usingderivatives of exponential functionsand thechain ruleto obtain
Proof of Power Rule 2
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