Mathematics
Medicine
The Derived Function
2010-2011
The Derived Function
Definition
Given a function y = f(x).
f(x) is said to have a derived function at x if and only if the following limit exists
and is finite;
the function
f ' ( x)
defined by the limit is called the derived function of f(x):
f ' ( x)  lim
h0
f ( x  h)  f ( x )
h
The derived function is also called the derivative of f(x) and the process of
finding the derivative is called differentiation. A function f(x) which has a
derivative at a point x = a is said to be differentiable at x = a.
Picture
y=f(x)
Y
Secant line
D
f(a+h)-f(a)
B
E
A
C
x=a
x=a+h
Tangent line
X
Geometric Interpretation of the Derived
Function
Let
y  f (x) have a graph (see the picture).
Let x = a be a point for which
f (a ) exists and let f ' ( x) exists.
The y coordinate of point B is f (a )
Let h be a real number, either positive or negative.
Then a + h is a point on the x axis and f ( x  h) is the y coordinate of point D.
Consequently, f (a  h)  f (a )
is difference in the y coordinates of points D and B
and may be positive or negative.
Its absolute value is depicted in the diagram as DE, the difference in the lengths of
CD and
AB
h
Notice that the length of the segment BE is
.
The ratio f (a  h)  f (a)
h
is slope of the line determined by B and D.
Such a line is a secant line.
Now think of h taking on values close to 0. For each h the ratio
f ( a  h)  f ( a )
h
is the slope of the corresponding secant line.
When h is are very close to 0, the points B and D are each on the graph of y  f (x)
and are very close together
Consequently, for small h, the secant line through B and D is very close to the line
tangent to f (x ) at x = a
Since the secant lines through B approach the line tangent to f(x) at x = a, the
slopes of the secant lines approach the slope of the tangent line. Since the
slope of each secant line is given by
f ( a  h)  f ( a )
h
the slope of the tangent is given by
lim
h 0
f ( a  h)  f ( a )
h
That is by the derived function of f (x) evaluated at x = a
Geometric Interpretation of f ' (a)
f ' (a)
is the slope of the line tangent to y  f (x) at x = a
Rules for Finding f'(x)
The derived function for most functions can be written down by application of
simple rules.
Theorem 1:
If f ( x)  k , k a constant, then f ' ( x)  0
Theorem 2:
The derived function of the product of a constant k and a function f (x)
'
is the product of k and f ( x)
Theorem 3:
If f ( x)  x m where m is a positive integer, then f ' ( x)  mx m1
Rules
Theorem 4:
If f ( x)  f1 ( x)  f 2 ( x)  ...  f n ( x) and f n' ( x ) exists for all n then
f ' ( x)  f '1 ( x)  f 2 ( x)  ...  f n ( x)
'
'
Theorem 5:
If
f ( x)  a0 x n  a1 x n1  a2 x n2  ...  an1 x  an
then
f ' ( x)  na0 x n1  (n  1)a1 x n2  (n  2)a2 x n3  ...  an1
Rules
Theorem 6:
If
f ( x)  x
1
then
f ' ( x) 
then
f ' ( x)  
2 x
Theorem 7:
If
1
f ( x)   
 x
1
x2
Theorem 8:
If f ( x)  1 , m a positive integer,
m
x
x0
, then
f ' ( x) 
m
x m1
Rules
Theorem 9:
If
f ( x)  cos( x)
then
f ' ( x)   sin( x)
then
f ' ( x)  cos( x)
Theorem 10:
f ( x)  sin( x)
If
Theorem 11:
If
f ( x )  tg ( x )
then
f ' ( x) 
1
cos2 ( x )
Theorem 12:
If
f ( x )  ctg ( x )
'
then f ( x )  
1
sin 2 ( x )
Rules
Theorem 13:
If f ( x )  arcsin( x ) then
f ' ( x) 
1
1 x2
Theorem 14:
If
f ( x )  arccos( x )
then
f ' ( x)  
1
1 x2
Theorem 15:
If f ( x )  arctg ( x )
then
f ' ( x) 
1
1 x2
Theorem 16:
If f ( x )  arcctg ( x ) then
f ' ( x)  
1
1 x2
Rules
Theorem 17:
If
f ( x)  a x
then
f ' ( x )  a x ln a
Theorem 18:
f ( x)  e
If
x
then
f ' ( x)  e x
Theorem 19:
If
f ( x )  log a x
then
f ' ( x) 
1
x ln a
Theorem 20:
If f ( x )  ln x
then
f ' ( x) 
1
x
Rules
Theorem 11:
If
f ( x)  r ( x) s( x) where r and s are differentiable function, then
f ' ( x)  r ( x) s ' ( x)  s ( x) r ' ( x)
Theorem 12:
If f ( x) 
r ( x)
where r and s are differentiable function and s  0
s ( x)
'
'
r
(
x
)
s
(
x
)

r
(
x
)
s
( x)
'
f ( x) 
s 2 ( x)
, then
Rules
Theorem 13:
If y  f (u )
(means y  f u(x)
u  u (x)
are differentiable function, then
y yu
'
x
'
u
'
x
, where
f (u )
and u (x )