Step 1: evaluate f(x+h) and f(x)
We have
[tex]f(x+h) = -(x+h)^2+6(x+h)-5 = -(x^2+2xh+h^2)+6x+6h-5[/tex]
[tex]= -x^2-2xh-h^2+6x+6h-5[/tex]
And, of course,
[tex]f(x)=-x^2+6x-5[/tex]
Step 2: evaluate f(x+h)-f(x)
[tex]f(x+h)-f(x)=-x^2-2xh-h^2+6x+6h-5-(-x^2+6x-5)=-2xh-h^2+6h[/tex]
Step 3: evaluate (f(x+h)-f(x))/h
[tex]\dfrac{f(x+h)-f(x)}{h}=-2x-h+6[/tex]
Step 4: evaluate the limit of step 3 as h->0
[tex]f'(x) = \displaystyle \lim_{h\to 0} \dfrac{f(x+h)-f(x)}{h}=-2x+6[/tex]
So, we have
[tex]f'(1) = -2\cdot 1+6 = 4,\quad f'(2) = -2\cdot 2+6 = 2,\quad f'(3) = -2\cdot 3+6 = 0[/tex]
