# FIND DERIVATIVES OF RADICAL FUNCTIONS

Find derivatives of radical functions :

Here we are going to see how to find the derivatives of radical functions.

We use the formula given below to find the first derivative of radical function.

f(x)  =  √x

f'(x)  =  1/(2√x)

Let us look into some example problems to understand the above concept.

Example 1 :

Find the derivative of the following function

y = (x3 + 2x)  √x

Solution :

y  =  (x+ 2x)  √x

Since two x terms are multiplying, we have to use the product rule to find the derivative.

 u = x3 + 2xu'  =  3x2 + 2(1)  =  3x2 + 2 v = √xv'  =  1/2√x

Product rule :

d (uv)  =  uv' + vu'

=  (x+ 2x)1/2√x +  √x(3x2 + 2)

=  (x3/2√x + 2x/2√x)  +  3x2√x + 2 √x

=  (1/2)x(3-1/2) + x(1 - 1/2)  +  3x(2 + 1/2) + 2 √x

=  (1/2)x5/2 + x1/2  +  3x5/2 + 2 √x

=  [(1/2) + 3]x5/2 √x + 2 √x

=  (7/2) x5/2 + 3 √x

Example 2 :

Find the derivative of the following function

y =  (√x + 2x)/x2 - 1

Solution :

y =  (√x + 2x)/x2 - 1

In the above question, In both numerator and denominator we have x functions.

So, we have to use the quotient rule to find the derivative

Quotient rule :

d (u/v)  =  (v u' - uv')/ v2

u  =  √x + 2x

u' = (1/2 √x) + 2(1)  ==>  (1/2√x) + 2

v = x2 - 1

v' = 2x - 0  ==> 2x

=  [(x2 - 1) ((1/2√x) + 2)) - (√x + 2x) (2x)] / (x2 - 1)2

Example 3 :

Find the derivative of the following function

y =  √(2x4 + 2x - 1)

Solution :

Let t  =  2x4 + 2x - 1

y = √t

dy/dx  =  (dy/dt) (dt/dx)

dy/dt  =  1/2√t    dy/dx  =  8x3 + 2(1) - 0

=  8x3 + 2

dy/dx  =  (1/2√t) (8x3 + 2)

By applying the value of t, we get

dy/dx  =  (1/2√2x4 + 2x - 1) (8x3 + 2)

=  (4x3 + 1)/(2x4 + 2x - 1)

After having gone through the stuff given above, we hope that the students would have understood "Find derivatives of radical functions"

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