As you use the term #f(x)# I am assuming you are using early stage Calculus.
Set #y=3x-2x^2# ........................(1)
Increment #x# by the minute amount of #deltax#
The this will cause a minute change in y of #deltay#
So
#y+deltay=3(x+deltax)-2(x+deltax)^2#
#=> y+deltay=3x+3deltax-2(x^2+2xdeltax+(deltax)^2)#
#=> y+deltay=3x+3deltax-2x^2-4xdeltax-2(deltax)^2)#
#=> y+deltay=3x+3deltax-2x^2-4xdeltax-2(deltax)^2#....(2)
Subtract equation (1) from equation (2)
#" " y+deltay=3x+3deltax-2x^2-4xdeltax-2(deltax)^2#....(2)
#" "underline( y" "= 3x" " -2x^2color(white)(........................) -) ..(1)#
#" " deltay=0" "+3deltax+0-4xdeltax-2(deltax)^2#
Divide throughout by #deltax#
#(deltay)/(deltax)= 3-4x-2deltax#
#lim_(deltaato0)(deltay)/(deltax)= 3-4x- lim_(deltaxto0)(2deltax) #
#(dy)/(dx)=3-4x#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
At #x=-3" "->" " (dy)/(dx)=3-4(-3) = +15#
At #x=+4" "->" " (dy)/(dx)=3-4(+4) = -13#
So mean rate of change is #(15-13)/2= +1#
