Difference quotient example

Question

Compute and simplify the difference quotient 

 

 f(x+h)-f(x)

 _________  , h≠0.

        h

 

f(x)=7x2 +9x

Answer 1

Wherever you see x in the function, substitute that for  (x + h).

 

 

 

f(x + h) - f(x)

___________  =

         h

 

 

 

7(x + h)² + 9(x + h) - (7x² + 9x)

___________________________  =

                  h

 

 

 

7(x² + 2xh + h²) + 9x + 9h - 7x² - 9x

_______________________________ =

                       h

 

 

 

7x² + 14xh + 7h² + 9x + 9h - 7x² - 9x 

_________________________________  =

                          h

 

 

The x² and x terms cancel each other out being that the coefficients are the same and have opposite signs.

 

14xh + 7h² + 9h

______________  =

           h

 

 

Factor out the h in the numerator.

 

h(14x + 7h + 9)

______________  =

             h

 

 

h on numerator and denominator cancel each other out.

 

14x + 7h + 9

 

 

Now we can plug in h=0, being that h is no longer in the denominator.

 

14x + 9