Derivatives of Quadratic Polynomials
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Given:f(x) = 2x2 + 2x + 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
2(x+h)2+(x+h)+7 - (2x2+2x+7)
h
h→0

f ' (x) = lim
2(x2+xh+h2)+2(x+h)+7 - (2x2+2x+7)
h
h→0

f ' (x) = lim
2x2+xh+2h2+2x+2h+7 + x2-2x-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+