# Power Rule in Differentiation (Part 3)

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

## Formula

Previously, you've learned that

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

when *n* is an integer.

Power rule in differentiation (Part 1)

Power rule in differentiation (Part 2)

This is also true

when *n* is a rational number: a fraction.

## Proof

Set *n* = *a*/*b*.

Then *y* = *x*^{n} can be changed into *y* = *x*^{a/b}.

Raise both sides to the power of *b*.

Then *y*^{b} = *x*^{a}.

Differentiate both sides.

[*y*^{b}]' = *b*⋅*y*^{b - 1}⋅*y*'

[*x*^{a}]' = *a*⋅*x*^{a - 1}

Implicit differentiation

To make [*y*' = ...] form,

divide both sides by *b*⋅*y*^{b - 1}.

Put *x*^{a/b} into the denominator's *y*.

Expand the denominator's exponent.

Cancel *a* and -*a*. (dark gray terms)

Change *a*/*b* into *n*.

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

when *n* is a rational number.

## Example 1

The square root is the power of 1/2.

So *y* = *x*^{1/2}.

Rational exponents

Use the power rule to find *y*'.*y*' = (1/2)*x*^{1/2 - 1}

So *y*' = 1/2√*x*.

The derivative of √*x* is frequently used.

So it's good to remember that [√*x*]' = 1/2√*x*.

If your teacher wants you

to rationalize the denominator,

rationalize the denominator

by multiplying √*x*/√*x*.

## Example 2

Change the radical into exponent form.

Then *y* = *x*^{-3/4}.

Rational exponents

Use the power rule to find *y*'.*y*' = (-3/4)*x*^{-3/4 - 1}

So *y*' = -3/4*x*^{4}√*x*.

If your teacher wants you

to rationalize the denominator,

rationalize the denominator

by multiplying ^{4}√*x*/^{4}√*x*.

## Example 3

Change the radical into exponent form.

Then *y* = (2*x* - 1)^{4/3}.

Rational exponents

Use the power rule and chain rule to find *y*'.