Part 4/5:
By substituting (y) back into the equation, we achieve our final expression for (y').
Cautionary Notes on Logarithm Definitions
While logarithmic differentiation is a robust technique, it’s important to note the domain restrictions. The natural logarithm is not defined for non-positive values. Thus, if the function (f(x)) involves negative values, direct application of natural logs won't work.
To work around this limitation, we can consider the absolute value:
[
y = \ln |f(x)|
]
This alteration allows us to differentiate even when (f(x)) is negative, as we can separate it into two cases—one for (x > 0) and one for (x < 0)—ultimately leading to a consistent derivative expression.