To find the difference quotient [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] for the function [tex]\( f(x) = \frac{1}{x+6} \)[/tex], we will follow these steps:
1. Evaluate [tex]\( f(x + h) \)[/tex]:
Given [tex]\( f(x) = \frac{1}{x+6} \)[/tex], substitute [tex]\( x + h \)[/tex] into the function:
[tex]\[
f(x + h) = \frac{1}{(x + h) + 6} = \frac{1}{x + h + 6}
\][/tex]
2. Form the difference quotient:
The difference quotient is given by:
[tex]\[
\frac{f(x + h) - f(x)}{h}
\][/tex]
Substitute [tex]\( f(x + h) \)[/tex] and [tex]\( f(x) \)[/tex] into this expression:
[tex]\[
\frac{\frac{1}{x + h + 6} - \frac{1}{x + 6}}{h}
\][/tex]
3. Combine the fractions in the numerator:
To combine the fractions, find a common denominator:
[tex]\[
\frac{1}{x + h + 6} - \frac{1}{x + 6} = \frac{(x + 6) - (x + h + 6)}{(x + h + 6)(x + 6)}
\][/tex]
Simplify the numerator:
[tex]\[
(x + 6) - (x + h + 6) = x + 6 - x - h - 6 = -h
\][/tex]
So, the numerator becomes:
[tex]\[
\frac{-h}{(x + h + 6)(x + 6)}
\][/tex]
4. Divide by [tex]\( h \)[/tex]:
Now, include the division by [tex]\( h \)[/tex]:
[tex]\[
\frac{\frac{-h}{(x + h + 6)(x + 6)}}{h} = \frac{-h}{h \cdot (x + h + 6)(x + 6)}
\][/tex]
5. Simplify the expression:
Notice that [tex]\( h \)[/tex] in the numerator and the denominator can be canceled:
[tex]\[
\frac{-h}{h \cdot (x + h + 6)(x + 6)} = \frac{-1}{(x + h + 6)(x + 6)}
\][/tex]
Therefore, the simplified difference quotient is:
[tex]\[
\frac{f(x+h)-f(x)}{h} = \frac{-1}{(x+6)(x+h+6)}
\][/tex]
This is the final simplified form.
