Final answer:
To simplify the difference quotient, substitute the function into the formula, expand and simplify the expression, and factor out h to get 10x + 5h - 1. Taking the limit as h approaches 0, the difference quotient becomes 10x - 1.
Step-by-step explanation:
To simplify the difference quotient for the function f(x) = 5x² - x - 6, we use the formula:
(f(x + h) - f(x)) / h
Substituting f(x) = 5x² - x - 6 into the formula, we get:
((5(x + h)² - (x + h) - 6) - (5x² - x - 6)) / h
Expanding and simplifying the expression, we have:
((5x² + 10hx + 5h² - x - h - 6) - 5x² + x + 6) / h
Simplifying further, we get:
(10hx + 5h² - h) /
Factoring out h, we get:
10x + 5h - 1
Taking the limit as h approaches 0, the difference quotient becomes 10x - 1.