The technique of logarithmic differentiation can be summarized in the following steps:

1.	Take the natural log of both sides, that is the log of f(x) on the left and the log of the function on the right. 
2.	Apply the identities for the log of a products or exponents to the right 
2.	Use the chain rule to take the derivative of both the left and right sides. The left side will always give you f '(x)/f(x).  the right side will either use the sum rule of differentiation or the product rule
3.	Multiply both sides by f(x) to isolate f '(x) on the left and an expression for f '(x) on the right. 
4.	Replace f(x) with the original product that you started with. 
5.	Optionally multiply the original product by the sum to simplify. 
Suppose you take the natural log of both sides of the equation 
   y=f(x)= ((x - 4)(x + 3)(x - 7)(x + 5) 
                 
   ln(f(x))  =  ln((x - 4)(x + 3)(x - 7)(x + 5))                  

Remember the property we have for the product of logs: 

   ln(f(x))  =  ln(x - 4) + ln(x + 3) + ln(x - 7) + ln(x + 5)     

Now apply the chain rule to the left side of equation and take the derivative, term by term, of the right side of equation:
  
    f '(x)       1           1         1           1
            =          +          +          +                              . 
     f(x)      x - 4      x + 3     x - 7     x + 5

From here it's easy to solve for f '(x). Just multiply through by f(x). Recall that 

f(x)= ((x - 4)(x + 3)(x - 7)(x + 5)
Simplify it by multiplying through. Notice that each fraction cancels a different factor in the original f(x). 
   f '(x)  =  (x + 3)(x - 7)(x + 5)  +  (x - 4)(x - 7)(x + 5)  +

             (x - 4)(x + 3)(x + 5)  +  (x - 4)(x + 3)(x - 7)