Derivative of a Radical
How to find the derivative of a radical function: formula (power rule), 2 examples, and their solutions.
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.
Change √x to the power of x: x1/2.
The derivative of x1/2, y', is,
write the exponent, 1/2
x1/2 - 1.
(1/2)x1/2 - 1 = (1/2)x-1/2
(1/2)x-1/2 = 1/(2x1/2)
x1/2 = √x
So y' = 1/(2√x) is the derivative of y = √x.
Change 1/4√x3 to the power of x: x-3/4.
The derivative of x-3/4, y', is,
write the exponent, -3/4
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 = 4√x3
So y' = -3/(4x⋅4√x3) is the derivative of y = 1/4√x3.