To find the derivative [tex]\( f'(x) \)[/tex] of the function [tex]\( f(x) = \frac{2x + 8}{5x + 5} \)[/tex]:
### Step-by-Step Solution
1. Identify the Function:
[tex]\[
f(x) = \frac{2x + 8}{5x + 5}
\][/tex]
2. Determine the Derivative Using the Quotient Rule:
The quotient rule states that if you have a function [tex]\( f(x) = \frac{g(x)}{h(x)} \)[/tex], then its derivative is:
[tex]\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
\][/tex]
Here, [tex]\( g(x) = 2x + 8 \)[/tex] and [tex]\( h(x) = 5x + 5 \)[/tex].
3. Compute [tex]\( g'(x) \)[/tex] and [tex]\( h'(x) \)[/tex]:
[tex]\[
g(x) = 2x + 8 \quad \Rightarrow \quad g'(x) = 2
\][/tex]
[tex]\[
h(x) = 5x + 5 \quad \Rightarrow \quad h'(x) = 5
\][/tex]
4. Apply the Quotient Rule:
[tex]\[
f'(x) = \frac{(2)(5x + 5) - (2x + 8)(5)}{(5x + 5)^2}
\][/tex]
5. Simplify the Expression:
- The numerator becomes:
[tex]\[
(2)(5x + 5) - (2x + 8)(5) = 10x + 10 - (10x + 40) = 10x + 10 - 10x - 40 = 10 - 40 = -30
\][/tex]
- Therefore:
[tex]\[
f'(x) = \frac{-30}{(5x + 5)^2}
\][/tex]
6. Simplified Derivative:
[tex]\[
f'(x) = \frac{-5(2x + 8)}{(5x + 5)^2} + \frac{2}{5x + 5}
\][/tex]
So the derivative of the function is:
[tex]\[
f'(x) = -\frac{5(2x + 8)}{(5x + 5)^2} + \frac{2}{5x + 5}
\][/tex]
### To find [tex]\( f'(4) \)[/tex]:
7. Substitute [tex]\( x = 4 \)[/tex] into the derivative:
[tex]\[
f'(4) = -\frac{5(2(4) + 8)}{(5(4) + 5)^2} + \frac{2}{5(4) + 5}
\][/tex]
Simplifying the terms:
[tex]\[
2(4) + 8 = 8 + 8 = 16
\][/tex]
[tex]\[
5(4) + 5 = 20 + 5 = 25
\][/tex]
So:
[tex]\[
f'(4) = -\frac{5 \cdot 16}{25^2} + \frac{2}{25}
\][/tex]
[tex]\[
f'(4) = -\frac{80}{625} + \frac{2}{25}
\][/tex]
Simplifying further:
[tex]\[
\frac{2}{25} = \frac{2 \times 25}{25 \times 25} = \frac{50}{625}
\][/tex]
[tex]\[
f'(4) = -\frac{80}{625} + \frac{50}{625} = \frac{-80 + 50}{625} = \frac{-30}{625} = -\frac{6}{125}
\][/tex]
So:
[tex]\[
f'(x) = -\frac{5(2x + 8)}{(5x + 5)^2} + \frac{2}{5x + 5}
\][/tex]
[tex]\[
f'(4) = -\frac{6}{125}
\][/tex]
