# Most of the Steps in a Logarithmic Differentiation Problem

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!

## 2 comments on “Most of the Steps in a Logarithmic Differentiation Problem” Add yours →

1. ChessBoozer says:

I see what you are saying. Student took derivative of left side, but not the right side of the equation.

2. Writing To Stay Sane says:

Thanks for putting up the problem on log derivatives, but could you put up one that has a few more steps and explanation. I promise I will donate quarter to your paypal if you write up something that helps me along in this bitch of calculus class we are taking at North Harris College!