Derivatives of Quadratic Polynomials
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Given:f(x) = 4x2 + 4x - 3
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)-3 - (4x2+4x-3)
h
h→0

f ' (x) = lim
4(x2+xh+h2)+4(x+h)-3 - (4x2+4x-3)
h
h→0

f ' (x) = lim
4x2+xh+4h2+4x+4h-3 + x2-4x+3
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+