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  yf (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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