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Question

For the function f(x)= 1/6x, construct and simplify the difference quotient f(x+h) - f(x)/h

Answer 1

The difference quotient = [f(x + h) - f(x)]/h

The function f(x) =1/6x

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

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

                    = [ 6x - (6x + 6h) ]/[(6x + 6h)6x]

                    = [ 6x - 6x - 6h]/[(6x + 6h)6x]

                    = - 6h/ (36x² + 36hx)

                    = - 6h/ 6(6x² + 6hx)

                    = - h/(6x² + 6hx)

The difference quotient = [- h/(6x² + 6hx)]/h

                                    = -1/(6x² + 6hx).