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
25. Differentiate \(g\left( x \right) = {\left( {\ln \left( {{x^2} + 1} \right) - {{\tan }^{ - 1}}\left( {6x} \right)} \right)^{10}}\) .
Show All Steps Hide All Steps
This problem will require multiple uses of the Chain Rule and so we’ll step though the derivative process to make each use clear.
Here is the first step of the derivative and we’ll need to use the Chain Rule in this step.
\[g'\left( x \right) = 10{\left( {\ln \left( {{x^2} + 1} \right) - {{\tan }^{ - 1}}\left( {6x} \right)} \right)^9}\frac{d}{{dx}}\left( {\ln \left( {{x^2} + 1} \right) - {{\tan }^{ - 1}}\left( {6x} \right)} \right)\] Show Step 2In this step we can see that we’ll need to use the Chain Rule on each of the terms.
The derivative is then,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{g'\left( x \right) = 10{{\left( {\ln \left( {{x^2} + 1} \right) - {{\tan }^{ - 1}}\left( {6x} \right)} \right)}^9}\left( {\frac{{2x}}{{{x^2} + 1}} - \frac{6}{{36{x^2} + 1}}} \right)}}\]