Derivatives of Quadratic Polynomials
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Given:f(x) = -2x2 + 9x - 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
-2(x+h)2+(x+h)-4 - (-2x2+9x-4)
h
h→0

f ' (x) = lim
-2(x2+xh+h2)+9(x+h)-4 - (-2x2+9x-4)
h
h→0

f ' (x) = lim
-2x2+xh-2h2+9x+9h-4 + x2-9x+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+