

The
derivative as a function 
The
derivative function 
Differentiation,
determining (or deriving) derivative of a function 
Derivatives
of basic or elementary functions 
Determining
the derivative of a function as the limit of the difference
quotient 
The
derivative of the power function 
The
derivative of the linear function

The
derivative of a constant

The
derivative of the sine function 
The
derivative of the cosine function 





The
derivative as a function 
The
derivative of a function f
(x)
is a new function that contains the value of the derivative of
all points on the original
function. 
That is, the
derivative function gives the slope of
the line tangent to f
(x)
at every point x. 
The
slope of the tangent line provides information about how the
graph of the function is changing. 
Thus,
if
f
'
(x)
<
0
then f
(x)
is decreasing 
and if f
'
(x)
>
0 then
f
(x)
is increasing, 
as is shown in the
figure
below. 


The
process of determining or finding the derivative is called differentiation. 
In
order a function to be differentiable, it must be continuous,
and both onesided limits (the
lefthanded and righthanded limits) must
be equal at
the given point. 
For example, the function f
(x)
=  x  is not differentiable
at x
= 0, although
it is continuous there. 

Derivatives
of basic or elementary functions

Determining
the derivative of a function as the limit of the difference
quotient 
We
use the limit definition 

to
find the derivative of a function.


The
derivative of the power function 
Given
is the power function
f (x)
= x^{n}
where n
is a a positive integer. 
We
use the binomial theorem to evaluate f
( x
+ h),


So
that, f
(x
+ h) 
f (x)
equals, 

Then
the difference quotient 

Since
every term except the first is factor of h
then,
the limit of
the difference quotient
as h
tends to zero is



Therefore,
if 

then 




The
derivative of the linear function

Thus,
for n = 1,
that is for the linear function f
(x)
= x, the
difference
f
(x
+ h) 
f (x) = x
+ h  x
= h so that the
difference quotient equals 1.


Therefore,
if

f(x)
= x 
then 




The
derivative of a constant

The
function f
(x)
= c,
where c
is a fixed constant, is graphically represented by a horizontal
line so that at any
given point (x,
f (x)) the slope
of the line tangent to the graph of f
is 0. Therefore, 
f(x)
= c and
f(x
+ h)
= c
so
that f(x
+ h) 
f(x)
= c  c
= 0. 
Thus, 


that
is, if

f(x)
= c 
then 




The
derivative of the sine function 
We
use the limit of the
difference quotient to find the
derivative of the function f
(x)
= sin x. 
Let
rewrite the
difference quotient applying the sum to product formula,


Since,
the derivative is the limit of the difference quotient as h
tends to zero then, 


Therefore,
if

f(x)
= sin x 
then 




The
derivative of the cosine function 
We
use the limit of the
difference quotient to find the
derivative of the function f
(x)
= cos x. 
Let
rewrite the
difference quotient applying the sum to product formula,


Since
the derivative is the limit of the difference quotient as h
tends to zero then, 


Therefore,
if

f(x)
= cos x 
then 











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contents F 



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