Question

Using the limit definition, how do you differentiate `f(x)=4 -2x -x^2`?

Answer 1

`f'(x) = -2-2x`

Explanation:

The limit definition of the derivative states that:

`f'(x) = lim_(Deltax->0) (f(x+Deltax)-f(x))/(Deltax)`

For `f(x)= 2-2x-x^2` we have:

`f'(x) = lim_(Deltax->0) ((2-2(x+Deltax)-(x+Deltax)^2)-(2-2x-x^2))/(Deltax)`

`f'(x) = lim_(Deltax->0) ((2-2x-2Deltax-x^2-2xDeltax-(Deltax)^2-2+2x+x^2))/(Deltax)`

`f'(x) = lim_(Deltax->0) ((-2Deltax-2xDeltax+(Deltax)^2))/(Deltax)`

`f'(x) = lim_(Deltax->0) (-2-2x+Deltax)=-2-2x`