Derivative of a Radical

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

Formula

Derivative of xn: 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: x1/2.

Rational Exponent

The derivative of x1/2, y', is,
write the exponent, 1/2
times
x1/2 - 1.

(1/2)x1/2 - 1 = (1/2)x-1/2

(1/2)x-1/2 = 1/(2x1/2)

Negative Exponent

x1/2 = √x

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

Example 2

Example

Solution

Change 1/4x3 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/x3/4 + 1

x3/4 + 1 = x1⋅x3/4

Product of Powers

x1 = x
x3/4 = 4x3

So y' = -3/(4x⋅4x3) is the derivative of y = 1/4x3.