Question 1202783
<font color=black size=3>
Answer: <font color=red size=4>-4x-2h-1</font>



Work Shown:


We first compute f(x+h)


{{{f(x) = -2x^2-x+3}}}


{{{f(x+h) = -2(x+h)^2-(x+h)+3}}} Each x replaced with (x+h).


{{{f(x+h) = -2(x+h)(x+h)-x-h+3}}}


{{{f(x+h) = -2(x^2+2xh+h^2)-x-h+3}}} FOIL rule


{{{f(x+h) = -2x^2-4xh-2h^2-x-h+3}}}


Now subtract off f(x)


{{{f(x+h)-f(x) = (-2x^2-4xh-2h^2-x-h+3) - (-2x^2-x+3)}}}


{{{f(x+h)-f(x) = -2x^2-4xh-2h^2-x-h+3 + 2x^2+x-3}}}


{{{f(x+h)-f(x) = -4xh-2h^2-h}}}


{{{f(x+h)-f(x) = h(-4x-2h-1)}}}


The last step is to divide over h.


{{{(f(x+h)-f(x))/h = (h(-4x-2h-1))/h}}}


{{{(f(x+h)-f(x))/h = (highlight(h)(-4x-2h-1))/(highlight(h))}}}


{{{(f(x+h)-f(x))/h = (cross(h)(-4x-2h-1))/(cross(h))}}}


{{{(f(x+h)-f(x))/h = -4x-2h-1}}}


Note: As h approaches 0, then {{{-4x-2h-1}}} approaches {{{-4x-1}}} which is the derivative of {{{f(x) = -2x^2-x+3}}}.
</font>