Question

Given `f(x) = 3x^2+6x`, what is `(f(x+h)-f(x))/h` ?

Answer 1

`(f(x+h)-f(x))/h = 6x+6+3h`

Explanation:

`f(x) = 3x^2+6x`

`f(x+h) = 3(x+h)^2+6(x+h)`

`color(white)(f(x+h)) = 3(x^2+2hx+h^2)+6(x+h)`

`color(white)(f(x+h)) = (3x^2+6hx+3h^2)+(6x+6h)`

`color(white)(f(x+h)) = (3x^2+6x) + 6hx+3h^2+6h`

`color(white)(f(x+h)) = f(x) + 6hx+6h+3h^2`

So:

`(f(x+h)-f(x))/h = (6hx+6h+3h^2)/h = 6x+6+3h`