Product Rule a,b are constants; f,g are functions; y , f , g denote derivatives.
Function
Derivative
y =af g
y =af g+af g
y = f g
n
y = f n gm
Product Rule
y = n (f g )n1 f g + f g )
Product-Chain Rule
y = n f f n1 g + m f g g m1
Chain-Product Rule
Ex1a.
Fractions and Rational Expressions
Algebraic Rules for Fractions and Rational Expressions
a, b, c, d may be numbers or variable expressions.
a c a+c
+=
bb
b
Adding and Subtracting
requires
a d c b ad + cd
ac
+=+=
bd
bd db
bd
a Common Denominator
ac
ac
=
b
Formulas for Exponent and Radicals
Algebraic Rules for Manipulating Exponential and Radicals Expressions.
In the following, n, m, k, j are arbitrary .
they can be integers or rationals or real numbers.
bn bm
= bn+mk
k
b
an bm
ck
j
=
anj bmj
ckj
Add expone
Handout Exercises - More Exponent and Radical Expressions
NOTE - there will be material from the previous quiz.
Remark: Recall that fractions should be combined into one fraction and
reduced. Often radicals need to be combined into one radical and reduced
0
What are and
?
0
Answer: M.W. = MORE WORK The problem is NOT DONE .
The more work is often Algebra!
These expressions mean More Work.
0
0
0
0
0
You should know the following (# denotes a non-zero number)
When you Plug-in and get these - you are done.
a)
Factoring Polynomials
Common Products
(U + V )2 = U 2 + 2U V + V 2
Squaring a Sum
(U V )2 = U 2 2U V + V 2
Squaring a Dierence
U 2 V 2 = (U + V )(U V )
Dierence of Two Squares
(U + V )3 = U 3 + 3U 2 V + 3U V 2 + V 3
Cubing a Sum
(U V )3 = U 3 3U 2 V + 3U
Handout - More Polynomials
Multiply the polynomials
a) (x 8)(3x + 4) x2 2x
c) 3 x 6
3 x+1
b) x3 + 1
d) x7/2 + 9
4x7/2 1
x3 + 3
5x4 + 6x3
3x3/2 5 x
5x9/2 9x7/2
Multiply and Add the polynomials
a) x12 4x11 x10 + 7x6 x5
b) 8x4 8x3 3x2 + 9x 8x2
c) x3/2 5 x
d)
Handout - Derivative - Power Rule
Power - First Rules
a,b are constants.
Function
Derivative
y = f (x)
dy
= f (x)
dx
dy
dx
x=#
Notation
= f (#)
Means Plug #
into derivative
y = a xn
dy
= a n xn1
dx
Power Rule
y =ax
dy
=a
dx
n = 1 in power rule
y=a
dy
=0
d
Handout - Derivative - Power Rule II
Calculate the derivative and Evaluate at the indicated value of x.
a) Evaluate f (3) for f (x) = 13x5
b) Evaluate f (2) for f (x) = 10x3
c) Evaluate f (3) for f (x) = 2x4
d) Evaluate f (1) for f (x) = 5x3
e) Evaluate f
Handout - Derivative - Chain Rule
Power-Chain Rule a,b are constants.
Function
Derivative
y = a xn
dy
= a n xn1
dx
Power Rule
y = a un
dy
du
= a n un1
dx
dx
Power-Chain Rule
Ex1a. Find the derivative of y = 8(6x + 21)8
Answer: y = 384(6x + 21)7
a = 8, n
Handout - Derivative - Chain Rule
&
Sin(x), Cos(x), ex , ln(x)
Power-Chain Rule a,b are constants.
Function
Derivative
y = a xn
dy
= a n xn1
dx
Power Rule
y = a un
dy
du
= a n un1
dx
dx
Power-Chain Rule
1
Exercises
Find the derivatives of the expressions
Handout - Fractions and Rational Expressions
Algebraic Rules for Fractions and Rational Expressions
a, b, c, d may be numbers or variable expressions.
a c a+c
+=
bb
b
Adding and Subtracting
requires
has Common Denominator
a d c b ad+cd
ac
+=+=
bd
bd db
bd