
The chain rule
applications 
Logarithmic differentiation 
Logarithmic differentiation examples 
Derivative of a composite exponential function 
Use of the logarithmic differentiation 
Derivatives of composite functions examples 





Logarithmic differentiation 
The derivative of the logarithm of
a function y
= f
(x)
is called the logarithmic derivative of the
function, thus


Therefore, the logarithmic derivative
is the derivative of the logarithm of a given function. 

Logarithmic differentiation examples 
Example:
Find
the derivative of the function f
(x)
= ln (sin x).

Solution: 



Example:
Find
the derivative of the function f
(x)
= ln (cos x).

Solution: 



Example:
Find
the
logarithmic derivative
of the function 



Solution:
Since the
logarithm of the given function 


then
differentiating both the left and the right side of the above
expression, obtained is 


Derivative
of a composite exponential function 
We
use the logarithmic differentiation to find derivative of a composite
exponential function of the form, 

where
u
and v
are functions of the variable x
and u
> 0. 
By
taking logarithms of both sides of the given exponential expression we
obtain, 
ln
y
= v
ln
u. 
Differentiating both sides of the above
equation
with respect to x 



Use of the logarithmic differentiation 
Derivatives of composite functions examples 
Example:
Find
the
derivative
of the function 



Solution: 


by
differentiating both sides of the above equation we get


or 



Example:
Find
the
derivative
of the function 



Solution: 


by
differentiating both sides of the above equation we get



Example:
Find
the
derivative
of the function 



Solution: 


by
differentiating both sides of the right equation above, we get 


Example:
Find
the
derivative
of the function 



Solution: 


by
differentiating both sides of the above equation we get











Calculus contents
C 



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