Sure! Let's start by finding the derivative [tex]\( f'(x) \)[/tex] of the function [tex]\( f(x) = \frac{6x + 4}{3x + 6} \)[/tex].
### Step 1: Find [tex]\( f'(x) \)[/tex] using the Quotient Rule
The Quotient Rule states that if you have a function [tex]\( f(x) \)[/tex] in the form [tex]\( \frac{u(x)}{v(x)} \)[/tex], then the derivative [tex]\( f'(x) \)[/tex] is given by:
[tex]\[
f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\][/tex]
where [tex]\( u(x) = 6x + 4 \)[/tex] and [tex]\( v(x) = 3x + 6 \)[/tex].
First, let's find the derivatives of [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex]:
[tex]\[
u'(x) = \frac{d}{dx}(6x + 4) = 6
\][/tex]
[tex]\[
v'(x) = \frac{d}{dx}(3x + 6) = 3
\][/tex]
Now, apply the Quotient Rule:
[tex]\[
f'(x) = \frac{(6)(3x + 6) - (6x + 4)(3)}{(3x + 6)^2}
\][/tex]
Simplify the numerator:
[tex]\[
= \frac{18x + 36 - (18x + 12)}{(3x + 6)^2}
\][/tex]
[tex]\[
= \frac{18x + 36 - 18x - 12}{(3x + 6)^2}
\][/tex]
[tex]\[
= \frac{24}{(3x + 6)^2}
\][/tex]
To further simplify:
[tex]\[
= \frac{6 \cdot 4}{(3x + 6)^2} = \frac{6}{3x + 6} - \frac{18 (6 x + 4)}{(3x + 6)^2} .
\][/tex]
Combining both, we use the result:
[tex]\[
f'(x) = 6/(3x + 6) - 3*(6x + 4)/(3x + 6)^2.
\][/tex]
### Step 2: Evaluate [tex]\( f'(x) \)[/tex] at [tex]\( x = 4 \)[/tex]
To find [tex]\( f'(4) \)[/tex], substitute [tex]\( x = 4 \)[/tex] into [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(4) = 6/(3(4) + 6) - 3*(6(4) + 4)/(3(4) + 6)^2
\][/tex]
[tex]\[
= 6/(12 + 6) - 3*(24 + 4)/(18)^2
\][/tex]
[tex]\[
= 6/18 - 3*(28)/324
\][/tex]
[tex]\[
= 1/3 - (84/324)
\][/tex]
[tex]\[
= 1/3 - 0.25925
\][/tex]
[tex]\[
f'(4) = 2/27
\][/tex]
Thus, the solution to the problem is:
[tex]\[
f'(x) = 6/(3x + 6) - 3*(6x + 4)/(3x + 6)^2
\][/tex]
[tex]\[
f'(4) = 2/27
\][/tex]
