As you use the term f(x)f(x) I am assuming you are using early stage Calculus.
Set y=3x-2x^2y=3x−2x2 ........................(1)
Increment xx by the minute amount of deltaxδx
The this will cause a minute change in y of deltayδy
So
y+deltay=3(x+deltax)-2(x+deltax)^2y+δy=3(x+δx)−2(x+δx)2
=> y+deltay=3x+3deltax-2(x^2+2xdeltax+(deltax)^2)⇒y+δy=3x+3δx−2(x2+2xδx+(δx)2)
=> y+deltay=3x+3deltax-2x^2-4xdeltax-2(deltax)^2)⇒y+δy=3x+3δx−2x2−4xδx−2(δx)2)
=> y+deltay=3x+3deltax-2x^2-4xdeltax-2(deltax)^2⇒y+δy=3x+3δx−2x2−4xδx−2(δx)2....(2)
Subtract equation (1) from equation (2)
" " y+deltay=3x+3deltax-2x^2-4xdeltax-2(deltax)^2 y+δy=3x+3δx−2x2−4xδx−2(δx)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
