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 (x 1 ) f (x) – represents (y 1 ) x + h– represents (x 2 ) f (x+h) – represents (y 2 ) f (x+h) – f (x)– represents (y 2 – y 1 ) h – represents (x 2 – x 1 ) Lim – represents the slope M as h→0

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

3. ► Create numerator f(x+h) – f(x) ► Remove brackets / combine like terms 3h 2 + 6xh +6h ► Combine numerator and denominator f(x+h) – f(x) = 3h 2 + 6xh + 6h hh f(x+h) {3x 2 + 6x – 4 + 3h 2 + 6xh +6h} – f(x) = – {3x 2 + 6x – 4} f(x+h) – f(x) = Note: You should have only “h” terms left in the numerator

4. f(x+h) – f(x) = 3h 2 + 6xh + 6h h ► Factor out common h f(x+h) – f(x) = h(3h + 6x + 6) 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

5. f ’(x) = lim f(x+h) – f(x) h→0 h Then f(x+h) – f(x) = h 0 3h + 6x + 6 6x + 6 f’(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