1. 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

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

3. f(x+h) – f(x) = 3h2 + 6xh + 6h h h
Note: You should have only “h” terms left in the numerator
►Remove brackets / combine like terms
►Combine numerator and denominator
►Create numerator f(x+h) – f(x)
f(x+h)
– f(x) =
{3x2 + 6x – 4 + 3h2+ 6xh +6h}
f(x+h) – f(x) =
3h2+ 6xh +6h
– {3x2 + 6x – 4}
f(x+h) – f(x) = 3h2 + 6xh + 6h
h h

4. ►Cancel h top and bottom
►Factor out common h
f(x+h) – f(x) = (3h + 6x + 6) h
f(x+h) – f(x) = h(3h + 6x + 6)
h h
f(x+h) – f(x) = 3h2 + 6xh + 6h
h h
f(x+h) – f(x) = (3h + 6x + 6) h

5. f’(x) = 6x + 6 f ’(x) = lim f(x+h) – f(x) h→0 h Then
If you are evaluating the limit of the equation as h goes to zero
f ’(x) = lim f(x+h) – f(x)
h→0 h
Then
Let ‘h’ go to zero
f(x+h) – f(x) = h
f’(x) =
6x + 6
3h + 6x + 6
3h + 6x + 6
6x + 6
f ’(x) represents the slope of the original equation at any x value.