Instances where Logarithmic Differentiation doesn't work?

[in the rye]

Hey,

In my class we just learned about logarithmic differentiation. I can see this being useful when taking the derivative of a complex function since it could be messy. But, I tried it on simpler equations as well. Everything I tried it on it seemed to work. Are there ever instances that it does not work?

To make sure we are using the same definition of logarithmic differentiation, I simply mean taking the log of both sides of an equation before taking its derivative. So where:

y = x²
ln y = ln (x²)
1 / y * y' = 1 / x² * 2x
y' = 2y / x
y' = 2x

Certainly just taking the derivative of this is easier, but it's just an example.

 

[Mark44]

Hey,

In my class we just learned about logarithmic differentiation. I can see this being useful when taking the derivative of a complex function since it could be messy. But, I tried it on simpler equations as well. Everything I tried it on it seemed to work. Are there ever instances that it does not work?

To the best of my knowledge, no.

To make sure we are using the same definition of logarithmic differentiation, I simply mean taking the log of both sides of an equation before taking its derivative. So where:

y = x²
ln y = ln (x²)
1 / y * y' = 1 / x² * 2x
y' = 2y / x
y' = 2x

Certainly just taking the derivative of this is easier, but it's just an example.

Your example is a very simple one, for which log differentiation is a lot more work than it would be by simpler means.

Here are some examples where log differentiation would be very useful.
1. ##y = (3x^2 + 5)^{1/x}##
2. ##y = (\sin x)^{x^3}##
(from https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/logdiffdirectory/LogDiff.html)