Paul's Online Math Notes
Calculus I (Notes) / Derivatives / Logarithmic Differentiation   [Notes] [Practice Problems] [Assignment Problems]

Calculus I - Notes

Internet Explorer 10 & 11 Users : If you have been using Internet Explorer 10 or 11 to view the site (or did at one point anyway) then you know that the equations were not properly placed on the pages unless you put IE into "Compatibility Mode". I beleive that I have partially figured out a way around that and have implimented the "fix" in the Algebra notes (not the practice/assignment problems yet). It's not perfect as some equations that are "inline" (i.e. equations that are in sentences as opposed to those on lines by themselves) are now shifted upwards or downwards slightly but it is better than it was.

If you wish to test this out please make sure the IE is not in Compatibility Mode and give it a test run in the Algebra notes. If you run into any problems please let me know. If things go well over the next week or two then I'll push the fix the full site. I'll also continue to see if I can get the inline equations to display properly.
Limits Previous Chapter   Next Chapter Applications of Derivatives
Higher Order Derivatives Previous Section   Next Section Applications of Derivatives (Introduction)

 Logarithmic Differentiation

There is one last topic to discuss in this section.  Taking the derivatives of some complicated functions can be simplified by using logarithms.  This is called logarithmic differentiation.


It’s easiest to see how this works in an example.


Example 1  Differentiate the function.



Differentiating this function could be done with a product rule and a quotient rule.  However, that would be a fairly messy process.  We can simplify things somewhat by taking logarithms of both sides.



Of course, this isn’t really simpler.  What we need to do is use the properties of logarithms to expand the right side as follows.



This doesn’t look all that simple.  However, the differentiation process will be simpler.  What we need to do at this point is differentiate both sides with respect to x.  Note that this is really implicit differentiation.



To finish the problem all that we need to do is multiply both sides by y and the plug in for y since we do know what that is.



Depending upon the person, doing this would probably be slightly easier than doing both the product and quotient rule.  The answer is almost definitely simpler than what we would have gotten using the product and quotient rule.


So, as the first example has shown we can use logarithmic differentiation to avoid using the product rule and/or quotient rule.


We can also use logarithmic differentiation to differentiation functions in the form.




Let’s take a quick look at a simple example of this.


Example 2  Differentiate  



We’ve seen two functions similar to this at this point.



Neither of these two will work here since both require either the base or the exponent to be a constant.  In this case both the base and the exponent are variables and so we have no way to differentiate this function using only known rules from previous sections.


With logarithmic differentiation we can do this however.  First take the logarithm of both sides as we did in the first example and use the logarithm properties to simplify things a little.




Differentiate both sides using implicit differentiation.



As with the first example multiply by y and substitute back in for y.



Let’s take a look at a more complicated example of this.



Example 3  Differentiate  



Now, this looks much more complicated than the previous example, but is in fact only slightly more complicated.  The process is pretty much identical so we first take the log of both sides and then simplify the right side.



Next, do some implicit differentiation.



Finally, solve for  and substitute back in for y.



A messy answer but there it is.


We’ll close this section out with a quick recap of all the various ways we’ve seen of differentiating functions with exponents.  It is important to not get all of these confused.





It is sometimes easy to get these various functions confused and use the wrong rule for differentiation.  Always remember that each rule has very specific rules for where the variable and constants must be.  For example, the Power Rule requires that the base be a variable and the exponent be a constant, while the exponential function requires exactly the opposite.


If you can keep straight all the rules you can’t go wrong with these.

Higher Order Derivatives Previous Section   Next Section Applications of Derivatives (Introduction)
Limits Previous Chapter   Next Chapter Applications of Derivatives

Calculus I (Notes) / Derivatives / Logarithmic Differentiation    [Notes] [Practice Problems] [Assignment Problems]

© 2003 - 2016 Paul Dawkins