# Difference Quotient (4 step method of slope)

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11-Jan-2016Category

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Difference Quotient (4 step method of slope)Also known as: (Definition of Limit), and (Increment definition of derivative)

f (x) = lim f(x+h) f(x) h0 hThis equation is essentially the old slope equation for a line: x represents (x1) f (x) represents (y1) x + h represents (x2) f (x+h) represents (y2)

f (x+h) f (x) represents (y2 y1) h represents (x2 x1)

Lim represents the slope M as h0

given substitute (x+h) for every x in f(x)f(x) = 3 x2 + 6 x 4 f(x+h) = 3(x+h)2 + 6(x+h) 4 expand (x+h)2f(x+h) = 3(x2 + 2xh + h2)+ 6(x+h) 4 remove parenthesesf(x+h) = 3x2 + 6xh + 3h2+ 6x+6h 4f(x+h) = 3x2 + 6x 4 + 3h2+ 6xh +6h combine like terms and organize Notice original f(x) in greenf(x+h) = 3x2 + 6x 4 + 3h2+ 6xh +6h

Create numerator f(x+h) f(x)Remove brackets / combine like terms3h2+ 6xh +6hCombine numerator and denominatorf(x+h) f(x) = 3h2 + 6xh + 6h hhf(x+h){3x2 + 6x 4 + 3h2+ 6xh +6h} f(x) = {3x2 + 6x 4} f(x+h) f(x) =Note: You should have only h terms left in the numerator

f(x+h) f(x) = 3h2 + 6xh + 6h hhFactor out common hf(x+h) f(x) = h(3h + 6x + 6) hhf(x+h) f(x) = (3h + 6x + 6) h 1Cancel h top and bottomf(x+h) f(x) = (3h + 6x + 6) h

f (x) = lim f(x+h) f(x) h0 hThenf(x+h) f(x) = h 03h + 6x + 63h + 6x + 66x + 66x + 6f(x) =f (x) represents the slope of the original equation at any x value.Let h go to zeroIf you are evaluating the limit of the equation as h goes to zero