I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 3.9 : Chain Rule
12. Differentiate \(V\left( x \right) = \ln \left( {\sin \left( x \right) - \cot \left( x \right)} \right)\) .
For this problem the outside function is (hopefully) clearly the logarithm and the inside function is the stuff inside of the logarithm. The derivative is then,
\[V^{\prime}\left( x \right) = \frac{1}{{\sin \left( x \right) - \cot \left( x \right)}}\left( {\cos \left( x \right) + {{\csc }^2}\left( x \right)} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{\cos \left( x \right) + {{\csc }^2}\left( x \right)}}{{\sin \left( x \right) - \cot \left( x \right)}}}}\]With logarithm problems remember that after differentiating the logarithm (i.e. the outside function) you need to substitute the inside function into the derivative. So, instead of getting just,
\[\frac{1}{x}\]we get the following (i.e. we plugged the inside function into the derivative),
\[\frac{1}{{\sin \left( x \right) - \cot \left( x \right)}}\]Then, we can’t forget of course to multiply by the derivative of the inside function.