

Derivatives
of basic or elementary functions 
Determining
the derivative of a function as the limit of the difference
quotient 
The
derivative of the exponential function 
The
derivative of the logarithmic function 
Differentiation
rules 
The
derivative of the sum or difference of two functions 
The
product rule 
A
constant times a function rule 
Derivatives of functions, examples 





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 




Differentiation
rules 
The
derivative of the sum or difference of two functions 
The
derivative of the sum or difference of two differentiable functions
equals the sum
or difference of their derivatives,
written 


The
product rule 
The
derivative of the product of two differentiable functions is
equal to, the first function times the derivative of the second
plus the second function times the derivative of the first, 


A
constant times a function rule 
The
derivative of a constant times a function is equal the constant
times the derivative of the function, where
the
constant c
can be any real number or expression that does not contain the
variable, 


Derivatives of
functions examples 
Example: 
Find
the derivative of the linear function f(x)
= m x
+ c. 

Solution: 




Example: 
Find
the derivative of the function 


Solution: 




Example: 
Find
the derivative of the function 


Solution: 




Example: 
Find
the derivative of the function 


Solution: 











Functions
contents F 



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