Final answer:
To find the difference quotient for the function (6)/(x), evaluate the function at x + h, set up the difference quotient formula, find the common denominator to combine fractions, and simplify by canceling out h in the numerator and denominator.
Step-by-step explanation:
To determine the difference quotient for the function (6)/(x), we typically start by considering the function at x and x + h, where h is a small increment to x. The difference quotient is then the expression (f(x + h) - f(x))/h. In this case, the function is f(x) = 6/x.
As a first step, we evaluate the function at x + h:
f(x + h) = 6/(x + h).
Next, we set up the difference quotient:
(f(x + h) - f(x))/h which simplifies to ((6/(x + h)) - (6/x))/h.
To compute this, we find a common denominator for the two fractions:
((6x - 6(x + h))/(x(x + h)))/h.
Expanding the numerator we get:
((6x - 6x - 6h)/(x(x + h)))/h which simplifies to -6h/(x(x + h)h).
Now, when we divide by h, we get:
-6/(x(x + h)), which is the difference quotient.