# Power rule in calculus

## Statement

The derivative of $x^r$ is $r * x^{r-1}$, where $r$ is a real number.

$$\tfrac{d}{dx}(x^r) = r * x^{r-1}, r \in \Reals$$

## Proof

Define function $f$.

$$f(x) = x^r$$

Rewrite the function using $e$ and $\ln$. Since they are inverse functions, it stays exactly the same.

$$f(x) = e^{\ln(x^r)}$$

Differentiate this function by the derivative of e^x, and mutliply by the chain rule.

$$f'(x) = e^{\ln(x^r)} * \tfrac{d}{dx}(\ln(x^r))$$

The first factor simplifies to $x^r$, since $e$ and $\ln$ are inverses.

$$f'(x) = x^r * \tfrac{d}{dx}(\ln(x^r))$$

For the second factor, bring down the exponent $r$ using the properties of logarithms. Then bring $r$ out of the $\frac{d}{dx}$ function.

$$f'(x) = x^r * r * \tfrac{d}{dx}(\ln(x))$$

The derivative of $\ln(x)$ is $\frac{1}{x}$.

$$f'(x) = x^r * r * \frac{1}{x}$$

Now simplify this function to the final result by using negative exponents and adding exponents.

$$f'(x) = r * x^r * x^{-1}$$

$$f'(x) = r * x^{r - 1}$$

## Proofs building upon this proof

### Exponent rule in calculus

This proofs show the derivative of a^x is a^x * ln(a).

### Quotient rule in calculus

This proof shows that the derivative for the quotient or fraction a/b is (a'b - ab') / b^2.

### The derivative of ln(x)

This proof shows that the derivative of ln(x) is 1/x.