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Math 1431
Section 12485 MWF 12‐1pm SR 117
Dr. Melahat Almus
almus@math.uh.edu
http://www.math.uh.edu/~almus
COURSE WEBSITE:
http://www.math.uh.edu/~almus/1431_sp16.html
Visit my website regularly for announcements and course material!
Read the syllabus (posted on the course website!).
If you e-mail me, please mention your course (1431) in the subject line.
Be considerate of others in class. Respect your friends and do not distract anyone during the lecture.
Check your CASA account for quiz due dates; don’t miss any quizzes.
Purchase popper scantrons from UH Bookstore. Bring one scantron to every class.
BUBBLE IN PS ID VERY CAREFULLY! If you make a bubbling mistake, your scantron will not be saved in the system and you will not get credit for it even if you turned it in.
Bubble in Popper Number.
2
Popper #
Question# If 3 22 5 2f x x x x , 1f ' ?
a) 5 b) 14 c) 12 d) 8 e) None
Question# If 3 2
25x x x
f xx
, 1f ' ?
a) 7 b) 5 c) 4 d) 2 e) None
Question# If 5 4f x cos x sin x , f ' ?
a) -5 b) 4 c) -4 d) 1 e) None
Question# If 6f x tan x cot x , 4
f ' ?
a) -10 b) 7 c) -12 d) 14 e) None
3
Recall--
Power Rule
If nf x x , where n is any positive integer, then 1nf ' x nx .
Slope of the tangent line at x a = Derivative of the function at x a .
tangentm f ' a and normal
tangent
1 1m
m f ' a .
Derivatives of the Six Trig Functions:
2
2
sin x ' cos x
cos x ' sin x
tan x ' sec x
cot x ' csc x
sec x ' sec x tan x
csc x ' csc x cot x
4
Higher Order Derivatives:
4 3 2 2 10f x x x x x
The first derivative of f : f ' x
The second derivative of f : 'f '' x f ' x
The third derivative of f : 'f ''' x f '' x
The fourth derivative of f : 4 'f x f ''' x
The fifth derivative of f : 5 4'
f x f x
In general, nf stands for the n th order derivative of f .
The functions f ' , f '' , f ''' , 4f ,…, nf are called the derivatives of f of
orders 1,2,3,…,n , respectively.
Remark:
4f x stands for the fourth order derivative of f x , while 4f x means
4f x .
To see a variant of this notation, let 5y x ;
45y' x , 320y'' x , 260y''' x , and so on.
5
With the double-d notation, the second order derivative is:
2
2d y d dy
dx dxdx
or 2
2d d d
f x f xdx dxdx
,
and the third order derivative is:
3 2
3 2d y d d y
dxdx dx
or
3 2
3 2d d d
f x f xdxdx dx
,
and so on.
Example: 4 2 5f x x x x ; 1f '' ?
6
Example: 5f x x x ; 2
2d f
?dx
Example: 10y cost ; 3
3d y
?dt
7
Section 2.3 – Differentiation Rules
Theorem: The Product Rule
If f and g are differentiable at x , then so is the product fg . Moreover,
fg ' x f ' x g x f x g' x
This formula may be written as:
uv ' u' v uv' or d du dvu v v u
dx dx dx .
This rule can be extended to the product of more functions:
uvw ' u' vw uv' w uvw' or
d du dv dwu v w v w u w u v
dx dx dx dx .
8
Example: Find the derivative of 3h x x cos x .
Example: If 4 22 5 6y x x x , dy
?dx
9
Theorem: The Reciprocal Rule
If f is differentiable at x and 0f x , then so is the reciprocal 1
f. Moreover,
2
1 f ' xd
dx f x f x
Example: For 21
g xx x
, find 2g' .
10
Theorem: The Quotient Rule
If f and g are differentiable at x and 0g x , then the quotient f / g is
differentiable at x and
2
'f ' x g x f x g' xf
xg g x
This formula may be written as:
2
'u u' v u v'
v v
.
Example: Find the derivative of 22
5
xf x
x
.
11
Example: Find the derivative of 5 2
sin xf x
x
.
Example: Find the slope of the tangent line to the curve 2
5
x xf x
x
at 1x .
12
Theorem: The Chain Rule
If g is differentiable at x and f is differentiable at g x , then the composition
f g is differentiable at x . Moreover,
f g ' x f ' g x g' x .
This rule is one of the most important rules of differentiation. It helps us with many complicated functions.
Example: Find the derivative of 32 1h x x .
13
Example: Find the derivative of 42 5h x x x .
Example: Find the derivative of 103 1h x x x .
14
POPPER #
Question# y sin x ,
3
3d y
?dx
a) cosx b) sinx c) –sinx d) –cosx e) None
Question# f x xcos x , f ' ?
a) 0 b) 1 c) -1 d) 2 e) None
Question# 2
xf x
x
, 2f ' ?
a) 1/2 b) 1/4 c) 1/8 d) 1/16 e) None
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