# Elementary power rule of derivatives/Proof

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, for is a natural number.

## Proof by Induction

### Prerequisites

- The product rule of derivatives,

### Proof

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.

QED

## Proof by Exponentiation

This proof is validates the power rule for all real numbers such that the derivative

### Prerequisites

- The exponential rule of derivatives,
- The chain rule of derivatives,

### Proof

## Proof by Binomial Expansion

This proof is only valid for positive real integer exponents,

### Prerequisites

- The Binomial Expansion:

- The first principle limit definition of the derivative