# Power Rule in Differentiation (Part 2)

How to solve the power rule in differentiation problems (when the exponent is a non-zero integer): formula, proof, example, and its solution.

## Formula

Previously, you've learned that

[*x*^{n}]' = *nx*^{n - 1}

when *n* is a natural number.

Power rule in differentiation (Part 1)

This is also true

when *n* is a non-zero integer: -1, -2, -3, ... .

## Proof

Set *n* = -*m*.

Then [*x*^{n}]' = [*x*^{-m}]'.

Differentiate 1/*x*^{m}.

Reciprocal rule in differentiation

Change -*m* to *n*.

Then [*x*^{n}]' = *nx*^{n - 1}.

## Example

Divide each term on the numerator by *x*^{3}.

Then *y* = 6*x*^{3} - 3 + 2*x*^{-1} - *x*^{-3}.

Use the power rule to find *y*'.*y*' = 6⋅3*x*^{2} - 0 + 2⋅(-1)*x*^{-2} - (-3)*x*^{-4}

Change the negative exponents into the fractions.

Then *y*' = 18*x*^{2} - 2/*x*^{2} + 3/*x*^{4}.