Difference Quotient 4 step method of slope Also

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

2 f(x) Notice substitute remove combine expand original parentheses (x+h) like (x+h) terms forinevery and green organize x in f(x) ► given 2 f(x+h) f(x) = 3 3(x+h) 3 xx 222++6 xh 2 xh ++ 6 6(x+h) +3 h xh 22)+ +– 6 x+6 h 6(x+h) 4 –– 44 f(x+h) = 3 x 2 + 6 x – 4 + 3 h 2+ 6 xh +6 h

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

►Cancel h top and bottom ►Factor out common h 2 ++ f(x+h) – f(x) = 3 h h(3 h 6 x ++ 6 h 6) 6 xh (3 h h 1 h

If you are evaluating the limit of the equation as h goes to zero f ’(x) = lim h→ 0 Then f(x+h) – f(x) h Let ‘h’ go to zero f(x+h) –f’(x) f(x) = = 6 x 3 h++66 x + 6 h f ’(x) represents 0 the slope of the original equation at any x value.