Question

F(x)=2x^3-7x+1 Find and simplify the difference quotient

Answer 1

The difference quotient is

$\dfrac{F(x+h)-F(x)}h=\dfrac{(2(x+h)^3-7(x+h)+1)-(2x^3-7x+1)}h$

Expand the numerator and you'll see cancellation of a factor of $h$ (assuming $h\neq0$):

$\dfrac{F(x+h)-F(x)}h=\dfrac{2x^3+6x^2h+6xh^2+2h^3-7x-7h+1-2x^3+7x-1}h$

$\dfrac{F(x+h)-F(x)}h=\dfrac{6x^2h+6xh^2+2h^3-7h}h$

$\dfrac{F(x+h)-F(x)}h=6x^2-7+6xh+2h^2$