Certainly! Let's find the derivative of the function [tex]\( f(x) = \frac{6x + 4}{3x + 6} \)[/tex] and then evaluate it at [tex]\( x = 4 \)[/tex].
### Step 1: Define the function
The function given is:
[tex]\[ f(x) = \frac{6x + 4}{3x + 6} \][/tex]
### Step 2: Find the derivative [tex]\( f'(x) \)[/tex]
To find the derivative of [tex]\( f(x) \)[/tex], we will use the quotient rule. The quotient rule states:
[tex]\[ \left( \frac{u(x)}{v(x)} \right)' = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2} \][/tex]
where [tex]\( u(x) = 6x + 4 \)[/tex] and [tex]\( v(x) = 3x + 6 \)[/tex].
First, calculate [tex]\( u'(x) \)[/tex] and [tex]\( v'(x) \)[/tex]:
[tex]\[ u'(x) = 6 \][/tex]
[tex]\[ v'(x) = 3 \][/tex]
Now, apply the quotient rule:
[tex]\[
f'(x) = \frac{(6)(3x + 6) - (6x + 4)(3)}{(3x + 6)^2}
\][/tex]
Simplify the numerator step-by-step:
[tex]\[
f'(x) = \frac{18x + 36 - (18x + 12)}{(3x + 6)^2}
\][/tex]
[tex]\[
f'(x) = \frac{18x + 36 - 18x - 12}{(3x + 6)^2}
\][/tex]
[tex]\[
f'(x) = \frac{36 - 12}{(3x + 6)^2}
\][/tex]
[tex]\[
f'(x) = \frac{24}{(3x + 6)^2}
\][/tex]
### Step 3: Evaluate [tex]\( f'(x) \)[/tex] at [tex]\( x = 4 \)[/tex]
Now, substitute [tex]\( x = 4 \)[/tex] into the derivative [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(4) = \frac{24}{(3 \cdot 4 + 6)^2}
\][/tex]
[tex]\[
f'(4) = \frac{24}{(12 + 6)^2}
\][/tex]
[tex]\[
f'(4) = \frac{24}{18^2}
\][/tex]
[tex]\[
f'(4) = \frac{24}{324}
\][/tex]
[tex]\[
f'(4) = \frac{2}{27}
\][/tex]
### Final Solution:
Thus, the derivative of the function [tex]\( f(x) = \frac{6x + 4}{3x + 6} \)[/tex] is
[tex]\[ f'(x) = \frac{24}{(3x + 6)^2} \][/tex]
And the value at [tex]\( x = 4 \)[/tex] is:
[tex]\[ f'(4) = \frac{2}{27} \][/tex]
So the solution is:
[tex]\[
\begin{array}{l}
f^{\prime}(x)=\frac{24}{(3 x+6)^2} \\
f^{\prime}(4)= \frac{2}{27}
\end{array}
\][/tex]
