, for is a natural number.
Proof by Induction
- The product rule of derivatives,
The proof proceeds by mathematical induction. Take the base case k=0. Then:
The induction hypothesis is that the rule is true for n=k:
We must now show that it is true for n=k+1:
Since the power rule is true for k=0 and given k is true, k+1 follows, the power rule is true for any natural number.
Proof by Exponentiation
This proof is validates the power rule for all real numbers such that the derivative
- The exponential rule of derivatives,
- The chain rule of derivatives,
Proof by Binomial Expansion
This proof is only valid for positive real integer exponents,
- The Binomial Expansion:
- The first principle limit definition of the derivative