Certainly! Let's find the difference quotient for the function [tex]\( f(x) = 6x - x^2 \)[/tex] and then simplify it.
First, we'll substitute [tex]\( x = 6 + h \)[/tex] into the function [tex]\( f(x) \)[/tex].
[tex]\[
f(6 + h) = 6(6 + h) - (6 + h)^2
\][/tex]
Simplify the expression inside the function:
[tex]\[
f(6 + h) = 36 + 6h - (36 + 12h + h^2)
\][/tex]
Distribute and combine like terms:
[tex]\[
f(6 + h) = 36 + 6h - 36 - 12h - h^2
\][/tex]
[tex]\[
f(6 + h) = 6h - 12h - h^2
\][/tex]
[tex]\[
f(6 + h) = -6h - h^2
\][/tex]
Next, calculate the value of [tex]\( f(6) \)[/tex]:
[tex]\[
f(6) = 6(6) - 6^2
\][/tex]
[tex]\[
f(6) = 36 - 36
\][/tex]
[tex]\[
f(6) = 0
\][/tex]
Now, compute the difference quotient:
[tex]\[
\frac{f(6+h) - f(6)}{h} = \frac{(-6h - h^2) - 0}{h}
\][/tex]
[tex]\[
\frac{f(6+h) - f(6)}{h} = \frac{-6h - h^2}{h}
\][/tex]
Simplify the expression by factoring and then canceling out [tex]\( h \)[/tex]:
[tex]\[
\frac{-6h - h^2}{h} = \frac{h(-6 - h)}{h}
\][/tex]
[tex]\[
\frac{h(-6 - h)}{h} = -6 - h
\][/tex]
So, the simplified difference quotient is:
[tex]\[
\boxed{-6 - h}
\][/tex]
