# What is the derivative of #f(x) = ln sqrt ((2x-8 )/( 3x+3))#?

##### 1 Answer

#### Explanation:

Let's do this. :)

First of all, let me simplify the expression using the following power rule:

#sqrt(x) = x^(1/2)#

and the following logarithmic rule:

#ln(a^r) = r * ln(a)#

Thus, we can simplify as follows:

# ln sqrt((2x-8)/(3x+3)) = ln [((2x-8)/(3x+3)) ^(1/2)] = 1/2 * ln ((2x-8)/(3x+3))#

This is easier already.

As next, we will need the chain rule:

#f(x) = 1/2 ln u " "# where#" " u = (2x-8)/(3x+3)#

Thus, the derivative of

Let's compute both:

#[1/2 ln u]' = 1/2 * 1 /u = 1 / 2 * 1 / ((2x-8)/(3x+3)) = 1 / 2 * (3x+3)/(2x-8)#

To compute the derivative of

for

#u = g/h# , the derivative is#u' = (g' h - h' g) / h^2#

In this case, we have

#u' = (2(3x+ 3 ) - 3(2x - 8)) / ((3x+3)^2) = 30 / (3x+3)^2#

In total, we have: