Answer:
Last option (counting from the top)
Step-by-step explanation:
For a given function f(x), the difference quotient is:
[tex]\frac{f(x + h) - f(x)}{h} = \frac{1}{h}*(f(x + h) - f(x))[/tex]
In this case, we have:
[tex]f(x) = \frac{8}{4x + 1}[/tex]
Then the difference quotient will be:
[tex]\frac{1}{h}*( \frac{8}{4*(x + h) + 1} - \frac{8}{4x + 1})[/tex]
Now we should get a common denominator.
We can do that by multiplying and dividing each fraction by the denominator of the other fraction, so we will get:
[tex]\frac{1}{h}*( \frac{8}{4*(x + h) + 1} - \frac{8}{4x + 1}) = \frac{1}{h}*(\frac{8*(4x + 1)}{(4(x + h) +1 )*(4x + 1)} - \frac{8*(4(x + h) + 1)}{(4(x + h) +1 )*(4x + 1)})[/tex]
Now we can simplify that to get:
[tex]\frac{1}{h}*\frac{8*(4x + 1) - 8*(4(x + h) + 1)}{(4(x + h) +1 )*(4x + 1)}} = \frac{1}{h}*\frac{-32h}{(4(x + h) +1 )*(4x + 1)}} = \frac{-32}{(4(x + h) +1 )*(4x + 1)}}[/tex]
Then the correct option is the last one (counting from the top)
