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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) h→0 h
This 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 h→0
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)2
f(x+h) = 3(x2 + 2xh + h2)+ 6(x+h) – 4
remove parentheses
f(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 green
f(x+h) = 3x2 + 6x – 4 + 3h2+ 6xh +6h
►
►Create numerator f(x+h) – f(x)►Remove brackets / combine like terms
3h2+ 6xh +6h
►Combine numerator and denominator
f(x+h) – f(x) = 3h2 + 6xh + 6h h h
f(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 h h
►Factor out common h
f(x+h) – f(x) = h(3h + 6x + 6) h h
f(x+h) – f(x) = (3h + 6x + 6) h 1
►Cancel h top and bottom
f(x+h) – f(x) = (3h + 6x + 6) h
f ’(x) = lim f(x+h) – f(x) h→0 h
Then
f(x+h) – f(x) = h
0
3h + 6x + 63h + 6x + 6
6x + 66x + 6f’(x) =f ’(x) represents the slope of the original equation at any x value.
Let ‘h’ go to zero
If you are evaluating the limit of the equation as h goes to zero