How do you know when to use logarithmic differentiation vs other differentiation rules. Provide a justification for which technique ...

Question

A peer asks you how do you know when to use logarithmic differentiation vs. other differentiation rules. Provide a justification for which technique to use. Include an example of each that illustrates when to use one technique over the other with the solutions explained and worked out.

Answer 1

usually logarithmic differentiation is great for when you have a variable in the base, and a variable in an exponent, like finding the derivative of y = x^x

finding the derivative of y = x^x would be impossible with chain rule, product rule or quotient rule, but easy with logarithmic differentiation.

To find the derivative of: y = x^x

take the logarithm of both sides, so you get ln(y) = ln(x^x)

Then use the properties of logarithms to rewrite as ln(y) = x*ln(x)

Then take the derivative of both sides: y'/y = lnx + 1

Then use algebra to get y' by itself: y' = y(lnx + 1)

Finally, substitute the original expression, which is equal to y: y' = x^x(lnx + 1)

I hope this helps!

Daniel W