Question

If `f(x)=x²-2x+3`, how do you find f(a+h)-f(a)/h?

Answer 1

Substitute `a+h` and `a` for `x` in the formula for `f(x)` and simplify to find:

`(f(a+h)-f(a))/h = 2a - 2 + h`

Explanation:

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

Then:

`(f(a+h) - f(a))/h`

`=(((a+h)^2-2(a+h)+3) - (a^2-2a+3))/h`

`=(color(red)(cancel(color(black)(a^2)))+2ah+h^2-color(blue)(cancel(color(black)(2a)))-2h+color(green)(cancel(color(black)(3)))-color(red)(cancel(color(black)(a^2)))+color(blue)(cancel(color(black)(2a)))-color(green)(cancel(color(black)(3))))/h`

`=2a-2+h`

So:

`lim_(h->0) (f(a+h) - f(a))/h = lim_(h->0) (2a-2+h) = 2a-2`

This is the derivative of `f(x)` at `x = a`