Question

What is the difference quotient `"f(x + h) - f(x)"/h`, where `h` is not equal to 0, of `f(x) = -4x^2 - 2x + 4`?

Answer 1

`-8x-4h-2`

Explanation:

Notice that the numerator contains the term `f(x+h)`. To find this, take `f(x)`, and plug in `x+h` wherever you see an `x`.

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

`f(x+h)=-4(x^2+2xh+h^2)-2x-2h+4`

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

Let's plug this back into the difference quotient with everything else.

`(stackrel"f(x+h)"overbrace(-4x^2-8xh-4h^2-2x-2h+4)-stackrel"f(x)"overbrace((-4x^2-2x+4)))/h`

`(cancel(-4x^2+4x^2)-8xh-4h^2cancel(-2x+2x)-2hcancel(+4-4))/h`

`(-8xh-4h^2-2h)/h`

`color(blue)(-8x-4h-2`