Derivatives of Quadratic Polynomials
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Given:f(x) = 4x2 - 5x - 8
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)-8 - (4x2-5x-8)
h
h→0

f ' (x) = lim
4(x2+xh+h2)-5(x+h)-8 - (4x2-5x-8)
h
h→0

f ' (x) = lim
4x2+xh+4h2-5x-5h-8 + x2+5x+8
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+