Chapter 2
The differential calculus and its applications
(for single variable)
The idea of derivation is first brought forward by French
mathematician Fermat.
The founders of calculus:
Englishman : Newton
German :Leibniz
derivative
differential
differential
Describe the speed of change of the function
Describe the degree of the change of the
function
2.1 Concept of derivatives
1.Introduction examples
2.Definition of derivatives
3.One-sided derivatives
4.The derivatives of some elementary functions
5.Geometric interpretation of derivative
6.Relationship between derivability and continuity
1.Introduction examples
Instantaneous velocity
Suppose that variation law of a moving
object is
Then the average velocity in the interval
( t，t0 ) is
f (t )  f (t0 )
v
t  t0
So the instantaneous
velocity at t 0 is
f (t )  f (t0 )
v  lim
t  t0
t t0
s
o
1 gt 2
2
f (t0 )
f (t )
t0
t
s
The slope of a tangent line to a plane curve
y
y  f (x )
the tangent line MT to
N
T
the limiting position of the
M
C
secant line M N
o   x0 x x
The slope of MT:
 lim tan 

f ( x)  f ( x0 )
The slope of MN: tan  
x  x0
f ( x)  f ( x0 )
k  lim
x  x0
x  x0
(when
)
Instantaneous velocity
f (t0 )
f (t )
t0
t
o
y
y  f (x )
N
Slope of the tangent
C
M
o   x0
The common character:
The limit of the quotient of the increments
Similar question:
acceleration
linear density
electric current

s
T
x x
2.Definition of derivatives
Definition2.1.1 .Suppose that
if
f ( x )  f ( x0 )

y
lim
 lim
x x0
x  x0
 x 0  x
is defined in
y  f ( x )  f ( x0 )
 x  x  x0
is said to be derivable at x0 ,and the limit is
exists, then
called the derivative of f at x0 ,denoted by
y  x  x0 ; f ( x0 ) ;
i.e.
y  x  x0
dy
d f ( x)
;
d x x  x0
d x x  x0
y
 f ( x0 )  lim
 x 0  x
y  f ( x )  f ( x0 )
 x  x  x0
note: If the limit above does not exist ,
then f is said to be non  derivable；
y
  , we say the derivation of
Especially, if lim
x 0  x
is infinite.
at
Exampl e 2. 1. 5 I nvest i gat e t he der i vabl i l i t y at x=0 of
1

x0
 xsin
f(x)= 
.
x
 0
x0
derived function of
f on I ;
Denoted by :
y ;
d y d f ( x)
.
;
f (x ) ;
dx
dx
note:
d f ( x0 )
f ( x0 )  f ( x) x  x0 
dx
3.One-sided derivatives
If the limit

( x  0 )
x0

( x  0 )
exists, then the limit is called the right (left) derivative of
denoted by f  ( x0 ) ( f  ( x0 ))
i.e.
f  ( x0 ) 


y
y x
For instance, for f ( x)  x
o
x
It is easy to know
f ( x 0 ) exists
f ( x0 )
Exampl e
sin x
I s t he f unct i on f(x)= 
x
x0
x0
der i vabl e at x=0？
Example2.1.3 prove the following formula for derivatives :
1 C   0;
 2   a x   a xlna;
(3)  e x   e x ;
(4)  x    x 1 (  );
(5)  sin x   cos x; (6)  cos x    sin x
1

(7)  ln x   ;
x
(8)  log a x  
1
x ln a
 lim f ( x  h)  f ( x)  lim a
h 0
h
h 0
a 1
x
 a lim

a
ln
a
h 0
h
h
x
 (a )  a ln a
x '
x
specially (e )  e
x '
x
xh
a
h
x

 1
(4)  x    x (  );

For instance，
( x ) 

(
1
( x 2 )
1
1

1
2

 x
2
2 x
1 
1

1
1

1
  ( x )   x
 2
x
x
1
x x
3
7



3
)  ( x 4 ) 
x 4
4
(5) for
f ( x  h)  f ( x )
sin( x  h)  sin x
 lim
 lim
h 0
h 0
h
h
h
 lim 2 cos( x  )
2
h 0
h
 lim cos( x  )
h 0
2
 cos x
(sin x)  cos x
f ( x  h)  f ( x )
log a ( x  h)  log a x
 lim
 lim
h 0
h
h 0
h
1
 lim 
h 0 h
 lim
x
1
h
1
x
h 0
lim
h
0
x
i.e.
1
(log a x ) 
x ln a
log a e
1
so (ln x ) 
x
'
P109 T2(2) Suppose that
f ( x0  h)  f ( x0  h)
lim
.
h 0
2h
exists, find the limit
Example
exists in
, find the value of a ,such that
and find
Solution:
sin x  0
f  (0)  lim 
1
x0
x 0
ax  0
a
f  (0)  lim 
x 0 x  0
and
So when a  1
5.Geometric interpretation of derivative
y
y  f (x )
'
f ( x0 )r epr sent s t he sl ope of t he t angent
l i ne t o t he cur ve y  f ( x ) at pi ont
( x0 , f ( x0 )).
i . e. tan   f ( x0 )
C
o 
T
M
x0
x
y
( x0 , y0 )
o
x0
 x
At the point ( x0 , f ( x0 ))
The tangent line:
The normal line:
( f ( x0 )  0 )
Exampl e2. 1. 6 Find the equations of the tan gen t
an d the normal to the curve y 
x at the po int (1,1).
6.Relationship between and continuity and derivabili
Th2.1.1.
Note：the converse is not necessarily true.
is derivable on [a , b]
Derivative
the rate of change
y
 aver age r at e of change.
x
y
f ( x0 )  lim  rate of change at a po int.
x 0 x
i nst ant aneous vel oci t y of a body movi ng al ong a st r ai ght
l i ne wi t h a var i abl e speed，
t he l i ne densi t y of a t hi n bar wi t h nonuni f or m mass di st r i but i on;
el ect r i c cur r ent ，
t he gr owt h r at e of a bi ol ogi cal popul at i on，
mar gi nal cost and so on ar e al l der i vat i ves .
Summarize:
1. The definition of derivative
f  ( x0 )  f ( x0 )  a
2. f ( x0 )  a
3. Geometric meaning: slope of the tangent line;
4. Derivable
continuous
whether continuous
5. How to judge the derivability by the definition
one-sided derivatives
6. Important derivatives :
(C )  0 ;
1
(log a x) 
x ln a
(cos x)   sin x ;
Have a think
and
difference: f (x)
relation:
is a function ,
f ( x) x  x0  f ( x0 )
？
attention:f ( x0 )  [ f ( x0 ) ]
f ( x0 ) is a value .
2. If
exists, then
f ( x0  h)  f ( x0 )  f ( x )
lim
 ________
0 .
h 0
h
3. We have
then
k0
Spare questions
1. Suppose
exists and
Solu:
1
f (1  ( x))  f (1)
 lim
2 x 0
( x)
so
find
P112 T1. Suppose
is continuous at
is derivable at
and
exists，
prove: