# Power Rule in Differentiation (Part 4)

How to solve the power rule in differentiation problems (when the exponent is a real number): formula, proofs, examples, and their solutions.

## Formula

Previously, you've learned that

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

when *n* is a rational number.

Power rule in differentiation (Part 3)

This is also true

when *n* is a real number.

## Proof

Set *y* = *x*^{n}

when *n* is a real number.

Then ln both sides,

covering the bases with the absolute value signs.

(The absolute value signs are added

becasue both sides can be (-).)

Natural logarithms

Differentiate both sides.

Implicit differentiation

Derivative of ln *x*

Multiply *y* on both sides.

Change *y* into *x*^{n}.

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

when *n* is a rational number.

## Example 1

*n* = √2

So *y*' = √2⋅*x*^{√2 - 1}.

## Example 2

*n* = *e*

So *y*' = *e*⋅*x*^{e - 1}.