Question

Use the four-step process to find f'(x) and then find f'(1), f'(2), and f'(3).

f(x) = -x^2+6x-5
f'(x) =

Answer 1

Step 1: evaluate f(x+h) and f(x)

We have

`f(x+h) = -(x+h)^2+6(x+h)-5 = -(x^2+2xh+h^2)+6x+6h-5`

`= -x^2-2xh-h^2+6x+6h-5`

And, of course,

`f(x)=-x^2+6x-5`

Step 2: evaluate f(x+h)-f(x)

`f(x+h)-f(x)=-x^2-2xh-h^2+6x+6h-5-(-x^2+6x-5)=-2xh-h^2+6h`

Step 3: evaluate (f(x+h)-f(x))/h

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

Step 4: evaluate the limit of step 3 as h->0

`f'(x) = \displaystyle \lim_(h\to 0) (f(x+h)-f(x))/(h)=-2x+6`

So, we have

`f'(1) = -2\cdot 1+6 = 4,\quad f'(2) = -2\cdot 2+6 = 2,\quad f'(3) = -2\cdot 3+6 = 0`