Given g(x) = -2x^2 + x + 6 determine: [g(x+h) - g(x)] / h

Task:  determine [g(x+h) - g(x)] / h

1.  Starting with g(x) = -2x^2 + x + 6, determine g(x+h):

    g(x+h) = -2(x+h)^2 + (x+h) + 6 = -2(x^2 + 2xh + h^2) + x + h + 6

                                                     =  

2.  Subtract g(x) from g(x+h):  

        g(x+h) - g(x) =   -2x^2 - 4xh -2 h^2 + x + h + 6

                              -   (2x^2 + x + 6                             )

                         =          - 4xh  - 2h^2 + h

3.  Divide this result by h:

     g(x+h) - g(x)

     =  -4x - 2h + 1                 (answer)

             h

Note:  Soon you will begin taking the limit (as h approaches 0) of such results.  Here that result would be -4x - 2(0) + 1 = -4x + 1.  This algebraic quantity is the "derivative" of the given function g(x) = -2x^2 + x + 6.

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