Let 𝑓(𝑥)=3𝑥^2 −4𝑥+2 Find 𝑓(𝑥+ℎ)−𝑓(𝑥) /ℎ and completely simplify your answer.

Answer 1

f(x) = 3x^2 -4x +2

f(x+h) = 3(x+h)^2 -4(x+h) +2 = 3(x^2+2xh+h^2) -4x -4h +2 = 3x^2 +6xh +3h^2 -4x -4h+2

f(x+h)-f(x) = 3x^2 +6xh +3h^2 -4x - 4h +2 - 3x^2 +4x -2 = 6xh+3h^2-4h

[f(x+h)-f(x)]/h = 6x +3h -4

as h approaches 0 [f(x+h)-f(x)]/h approaches 6x -4 = f'(x) = the derivative of f(x)