Taking the derivatives of some complicated functions can be simplified by using logarithms.  This is called logarithmic differentiation.
Suppose you wanted to take the derivative of

f(x)  =  (x - 4)(x + 3)(x - 7)(x + 5) 
                        
The function does not look too complicated. And there are strategies to get the derivative

For Instance: 
You could multiply out the polynomial and take the derivative of the result, but that's a lot of work.

Another strategy: 
You could observe that this is the product of  (x - 4)  and  (x + 3)(x - 7)(x + 5)  and use the product rule. But then you have to know the derivative of  (x + 3)(x - 7)(x + 5).  To find that you would have to observe that it, in turn, is the product of  (x + 3)  and  (x - 7)(x + 5)  and use the product rule on that. Etc.This is also turning out to be a lot of work . And it would be even worse if the original expression had more factors to it. 

This is an example where the strategy of logarithmic differentiation comes in handy. 

Another complicated situation function would be finding the derivative of functions of the nature:

 

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 
3.	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
4.	Multiply both sides by f(x) to isolate f'(x) on the left and an expression for f'(x) on the right. 
5.	Replace f(x) with the original product that you started with. 
6.	Optionally multiply the original product by the sum to simplify.