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.
