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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