Find the difference quotient fx h fxh where h notequalto 0

Find the difference quotient f(x + h) - f(x)/h, where h notequalto 0, for the function below. f(x) = 1/x + 6 Simplify your answer as much as possible.

Solution

f(x) = 1 / (x+6 )

f(x+h) - f(x) / h

f(x+h) = 1 / (x+h+6 )

f(x+h) - f(x) / h = {1 / (x+h+6) - 1 / (x+6)} / h

(x+6) - (x+h+6) / (x+h+6)(x+6) / h

= - 1 / (x+h+6) (x+6)