Let's find the derivative of the function [tex]\( f(x) = \frac{1}{4x + 1} \)[/tex] at [tex]\( x = 5 \)[/tex]. We'll proceed with the computation step-by-step.
### Step 1: Compute and Simplify [tex]\( f(5 + h) \)[/tex]
First, find the expression for [tex]\( f(5 + h) \)[/tex]:
[tex]\[ f(5 + h) = \frac{1}{4(5 + h) + 1} \][/tex]
Simplify the expression inside the function:
[tex]\[ f(5 + h) = \frac{1}{4 \cdot 5 + 4h + 1} = \frac{1}{20 + 4h + 1} = \frac{1}{4h + 21} \][/tex]
So,
[tex]\[ f(5 + h) = \frac{1}{4h + 21} \][/tex]
### Step 2: Compute and Simplify the Difference Quotient
Now, we need to compute the difference quotient:
[tex]\[ \frac{f(5 + h) - f(5)}{h} \][/tex]
First, calculate [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = \frac{1}{4 \cdot 5 + 1} = \frac{1}{20 + 1} = \frac{1}{21} \][/tex]
Substitute [tex]\( f(5 + h) \)[/tex] and [tex]\( f(5) \)[/tex] into the difference quotient:
[tex]\[ \frac{\frac{1}{4h + 21} - \frac{1}{21}}{h} \][/tex]
Combine the fractions in the numerator:
[tex]\[ \frac{1}{4h + 21} - \frac{1}{21} = \frac{21 - (4h + 21)}{21(4h + 21)} = \frac{21 - 4h - 21}{21(4h + 21)} = \frac{-4h}{21(4h + 21)} \][/tex]
The difference quotient becomes:
[tex]\[ \frac{\frac{-4h}{21(4h + 21)}}{h} = \frac{-4h}{21(4h + 21) \cdot h} = \frac{-4}{21(4h + 21)} \][/tex]
### Step 3: Take the Limit of the Difference Quotient as [tex]\( h \)[/tex] Approaches 0
Finally, we need to compute the limit of the difference quotient as [tex]\( h \)[/tex] approaches 0:
[tex]\[ f'(5) = \lim_{h \to 0} \frac{-4}{21(4h + 21)} \][/tex]
Substitute [tex]\( h = 0 \)[/tex] into the limit expression:
[tex]\[ f'(5) = \frac{-4}{21 \cdot 21} = \frac{-4}{441} \][/tex]
So, the derivative of the function at [tex]\( x = 5 \)[/tex] is:
[tex]\[ f'(5) = -\frac{4}{441} \][/tex]
