Logarithmic differentiation, many feel, simplifies the act of taking the derivative of a complicated product. Without it, one would have to use the product rule and the chain rule and then do some involved factoring to simplify the result.
Taking the logarithm of a product that involves powers simplifies the process initially due to the nice properties of logarithms that convert products to sums AND exponents(powers) to factors. This coupled with the easy formula for differentiating the logarithm of a function is why many prefer to use logarithmic differentiation.
That easy formula for the derivative of the natural log of a function is simply the derivative of the function divided by the function.
The following step by step process, by a student on a Calculus Final, actually left out a few of the steps. On the first step the student took the derivative of ln(y) but did not take the derivative of the right hand side. On the first two steps the right hand side was transformed via the laws of logarithms mentioned above.
The last 3 steps are correct and the student only left out some algebraic simplification at the end. The omitted work involved the cancellation of like binomial factors.
A hand writing analyst would have a field day with the small size of the students handwriting. I guess some analysts might conclude that the student has low self-esteem. It is rare that a student’s hand writing is smaller than the printed characters of the problem!