Derivatives of Quadratic Polynomials
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Given:f(x) = -4x2 - 11x + 7
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)+7 - (-4x2-11x+7)
h
h→0

f ' (x) = lim
-4(x2+xh+h2)-11(x+h)+7 - (-4x2-11x+7)
h
h→0

f ' (x) = lim
-4x2+xh-4h2-11x-11h+7 + x2+11x-7
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+