Product and Quotient Rules
1 The Product Rule:
If a function Is the product oftwo differentiable functions then the derivative' Is thd rst
times the derivative of the second plus the second times the derivative of the rst."
/ (2 cl when) W
(I
Emu -g(x)]=
DEFlNlTlON:
The line 3! = b is a horizontal asymptote of the graph of a function y = x)
it ith r.
e 8 ign;f(x)=b or xErme(x):b 3~(WK 93,17
+0~M
These uldelineeonl a l tolimlteatinnit so be careful.
HERE ARE THE RU LE6 TO FOLLOW:
l. if degree of num
mix will 3 NM mm Xa/
Basie Differentiation Rules
/\
l . The Constant Rule: 7) x)
The derivative of a constant function is O.
For any real number, 5: i [c] = 0
A.) y = -4 a.) s0) = 45 a) f(x) = 0 DJ 3 =3?
2. The Single Variable Rule:
The derivative of X
A Continuityand OneSided Limits
Definition of Continuity
Continuity atla Point: Fro erties of Continuit :
A function fis continuous at cif the following Given functions fand goontnuous at x = c,
three conditions are met: ,\)O\U then the following func
RELATIONSHIP OF f ' TO lNCREASING/DECREASING A
r l
When f>0,ther1fis MM;-
When f'<0,thenfisI cfw_Cm-5&5]
WHERE TO FIND EXTREMA
Relative (or Local Extrema (Maximums and Mnhnums) are values of the function that
are larger (max) or smaller (min) than the fun
The Fundamental Theorem ofCalculus
We have now seen the two major branches of calculus:
l) differential (tangent line problem)
2) integral (area problem)
Leibniz and Newton independently discovered a connection between them. stated informally, that
differ
THE GENERAL POWER RULE FOR INTEGRATION:
R
if g is a differentiable Function 0H: then
n+1
f 9001" g'(x)dx= 93311 + Cm. a 1
Equivaiently, 'rF u = g (x), then
un+1 .
71d = _
fu u n+1+C,n 1
if the function u: g (x) has a continuous derivative
1St DERIVATIVE TEST FOR EXTREMA
\ c
Using the Criticai Numbers (values of x where MACMM )
create a Sign chart and look to the left and right of these critical numbers
0R
Look at a graph of the derivative and see
When f '(x) changes signs om + to ,