Question

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

Answer 1

`f'(x) =1-4x`,
see the derivation below.

Explanation:

By definition, the derivative of the function `f(x)` is
`f'(x) = lim_(Delta x -> 0) (f(x+Delta x)-f(x))/(Delta x)`

In our case it looks like

`f'(x) =`
`= lim_(Delta x -> 0) (4+(x+Delta x)-2(x+Delta x)^2-4-x+2x^2)/(Delta x)=`

`= lim_(Delta x -> 0)(Delta x-4x Delta x-2(Delta x)^2)/(Delta x)=`

`= lim_(Delta x -> 0)(1-4x-2Delta x) =`

`= 1-4x - lim_(Delta x -> 0)(2Delta x) = 1-4x`