# Derivative of a Radical

How to find the derivative of a radical function: formula (power rule), 2 examples, and their solutions.

## Formula

### Derivative of x^{n}: Power Rule

To find the derivative of a radical,

change it to the power of x,

then use the power rule you've learned.

## Example 1

### Example

### Solution

Change √x to the power of x: x^{1/2}.

Rational Exponent

The derivative of x^{1/2}, y', is,

write the exponent, 1/2

times

x^{1/2 - 1}.

(1/2)x^{1/2 - 1} = (1/2)x^{-1/2}

(1/2)x^{-1/2} = 1/(2x^{1/2})

Negative Exponent

x^{1/2} = √x

So y' = 1/(2√x) is the derivative of y = √x.

## Example 2

### Example

### Solution

Change 1/^{4}√x^{3} to the power of x: x^{-3/4}.

The derivative of x^{-3/4}, y', is,

write the exponent, -3/4

times

x^{-3/4 - 1}.

x^{-3/4 - 1} = x^{-(3/4 + 1)} = 1/x^{3/4 + 1}

x^{3/4 + 1} = x^{1}⋅x^{3/4}

Product of Powers

x^{1} = x

x^{3/4} = ^{4}√x^{3}

So y' = -3/(4x⋅^{4}√x^{3}) is the derivative of y = 1/^{4}√x^{3}.