The derivative of a function f(x) (with respect to x) is defined as
color(white)("XXX")(df_x)/(dx)=lim_(hrarr0) (f(x+h)-f(x))/h
Given: f(x)=-3x^2+2x+9
color(white)("XXX")f(x+h)=-3(x+h)^2+2(x+h)+9
color(white)("XXXXXXXX")=-3x^2-3xh-3h^2+2x+2h+9
{:
(f(x+h)," = ",,-3x^2,-6xh,-3h^2,+2x,+2h,+9,),
(-f(x)," = " ,"- [ ",underline(-3x^2),underline(color(white)(_3xh)),underline(color(white)(-3h^2)),underline(+2x),underline(color(white)(+2h)),underline(+9)," ]"),
(f(x+h)-f(x)," = ",,,-6xh,-3h^2,,+2h,,)
:}
rArr color(white)("XX")(f(x+h)-f(x))/h=-6x-3h+2
and
color(white)("XXX")lim_(hrarr0) (f(x+h)-f(x))/h=-6x-3 * (0) + 2 = color(green)(-6x+2)
