Question

How to solve for the derivative given this information: `f(x+h)-f(x) = 2 h x^2 + 3 h x - 8 h^2 x + 8 h^2 - 4 h^3` ?

Answer 1

Given: `f(x+h)-f(x) = 2 h x^2 + 3 h x - 8 h^2 x + 8 h^2 - 4 h^3`

Divide both sides of the equation by h:

`(f(x+h)-f(x))/h = 2x^2 + 3x - 8 h x + 8 h - 4 h^2`

Compute the limit as `h to 0`

`lim_(h to 0) (f(x+h)-f(x))/h = 2x^2 + 3x`

This is the definition of the derivative.