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We want to get the difference quotient for the given function, we will get see that the difference quotient is equal to:
f'(x) = 2x + 5
For a given function f(x), we define the difference quotient as:
[tex]\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[/tex]
In this case, we have:
f(x) = x^2 + 5x + 6
Replacing that in the difference quotient we get:
[tex]\lim_{h \to 0} \frac{(x + h)^2 + 5*(x + h) + 6 - x^2 - 5x - 6}{h}\\\\\lim_{h \to 0} \frac{x^2 + 2xh + h^2 + 5*x + 5*h + 6 - x^2 - 5x - 6}{h}\\\\\lim_{h \to 0} \frac{ 2xh + h^2 + 5*h }{h}\\[/tex]
Now we can cancel the factor h to get:
[tex]\lim_{h \to 0} 2x + h + 5 = 2x + 0 + 5 = 2x + 5[/tex]
So the difference quotient is equal to 2x + 5.
If you want to learn more about difference quotients, you can read:
https://brainly.com/question/4224465