

Differential calculus  derivatives 
Derivatives
of basic or elementary functions 
The
derivative of the sine function 
The
derivative of the cosine function 
The
derivative of the exponential function 
The
derivative of the logarithmic function 






We
use the limit definition 

to
find the derivative of a function.


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 




The
derivative of the exponential function 
Let
use the limit of the
difference quotient to find the
derivative of the function f
(x)
= a^{x}. 
Since
the
difference quotient is


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

That
is, by plugging t
= a^{h} 
1, then t
®
0
as h ®
0,
and




Therefore,
if

f
(x)
= a^{x} 
then 



or
when the base a^{}
is substituted by the natural base e
obtained is the exponential function e^{x},
thus


if

f
(x)
= e^{x} 
then 




The
derivative of the logarithmic function 
Let's
use the limit of the
difference quotient to find the
derivative of the function f
(x)
= log_{a} x. 
The
difference quotient applied to the given function 

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


Then, applying the base
change identity and substituting a
= e 


if
f
(x)
= log_{a} x 
then 




if
f
(x)
= ln x 
then 











Calculus contents
C 



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