Implicit differentation has a very nice application which enables us to more easily find the derivative of functions involving large products, quotients or powers - and this is in fact the only useful method if the variable appears in the exponent and in the base (e.g. for \(f(x)=x^x\)).

This method is called “Logarithmic Differentiation”, and uses the special properties of logarithms to split products, quotients, and powers into simpler combinations of individual logarithms - and we often use this even if there was actually no logarithm in the given function, by manually introducing it!

Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

  • Review basic properties of logarithms.

  • Know the Chain Rule version of the derivative of the natural logarithm: \(\frac{d}{dx}\left[ \ln(f(x)) \right] = \frac{f'(x)}{f(x)}\).

  • Be able to derive functions of the form \(\ln(g(x))\) for simple functions \(g(x)\).

  • Review the method of implicit differentiation.

Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

  • Use properties of logarithms to split the logarithm of a large product, quotient, or power into a combination of individual logarithms.

  • Find the derivative of such a combination of individual logarithms.

  • Use logarithmic differentiation to find the derivative of functions which involve the variable in both the base and the exponent, e.g. \(f(x)=x^x\).

To prepare for class

  • Read section 0.4 to review basic properties of logarithms (in particular the overview titled “Properties of Logarithms”). Optionally, watch the videos and do the exercises/quiz questions on this page by the University of Waterloo, especially (but not only) lessons 3, 4, and 5: https://courseware.cemc.uwaterloo.ca/8/35/assignments/210/0

  • Do the Preview Activity about Logarithmic Differentiation on WeBWorK (if your teacher assigned it - there is no preview activity for this in Active Calculus).

  • Watch the following video about finding the derivative of functions involving logarithms:

  • Watch these videos to learn how we can apply a logarithm to simplify the process of finding the derivative of a function which actually did not have a logarithm itself (this is what is usually called “logarithmic differentiation”):

  • Optional: Review implicit differentiation (if necessary), at the very least by watching the following video:


Charles Fortin Avatar Charles Fortin
Gabriel Indurskis Avatar Gabriel Indurskis


Last Updated





Please click here if you find a mistake or broken link/video, or if you have any other suggestions to improve this page!