Question

Find f(x+h)-f(x)/h for the given function? f(x)=5x^2-6x+4

Answer 1

`(f(x + h) - f(x))/h = 5h + 10x - 6`

Explanation:

Quite simply, the value of `(f(x + h) - f(x))/h` is

`(5(x + h)^2 - 6(x + h) + 4 - (5x^2 - 6x + 4))/h`

`(5x^2 + 10xh + 5h^2 - 6x - 6h + 4 - 5x^2 + 6x - 4)/h`

`(5h^2 + 10xh - 6h)/h`

`(h(5h + 10x - 6))/h`

`5h + 10x - 6`

If you want to take this a step further, you can realize that the limit as `h-> 0` is the derivative, so

`f'(x) = lim_(h->0) 5h + 10x -6 = 10x - 6`

Hopefully this helps!

Answer 2

`10x+5h-6`

Explanation:

Let's first determine `f(x+h).` Finding this will entail replacing all `x` in `f(x)=5x^2-6x+4` with `x+h.`

`f(x+h)=5(x+h)^2-6(x+h)+4`
Expand and distribute:
`f(x+h)=5(x^2+2xh+h^2)-6x-6h+4`
`f(x+h)=5x^2+10xh+5h^2-6x-6h+4`

Thus,

`(f(x+h)-f(x))/h=(5x^2+10xh+5h^2-6x-6h+4-(5x^2-6x+4))/h=(cancel(5x^2)+10xh+5h^2cancel(-6x)-6hcancel(+4)cancel(-5x^2)cancel(+6x)cancel(-4))/h=(10xh+5h^2-6h)/h=(cancelh(10x+5h-6)/cancelh)=10x+5h-6`