Derivatives of Quadratic Polynomials
View the Lesson | MATHguide homepage Updated January 7th, 2022


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Given:f(x) = -4x2 - 8x - 4
Problem:Determine the derivative of f(x) using the definition of a derivative.

f ' (x) = lim
f(x+h) - f(x)
h
h→0

f ' (x) = lim
-4(x+h)2+(x+h)-4 - (-4x2-8x-4)
h
h→0

f ' (x) = lim
-4(x2+xh+h2)-8(x+h)-4 - (-4x2-8x-4)
h
h→0

f ' (x) = lim
-4x2+xh-4h2-8x-8h-4 + x2+8x+4
h
h→0

f ' (x) = lim
xh+h2+h
h
h→0

f ' (x) = lim
h(x+h+)
h
h→0

f ' (x) = limx+h+
h→0

f ' (x) =  x+