General Notice
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
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Section 3.9 : Chain Rule
13. Differentiate \(h\left( z \right) = \sin \left( {{z^6}} \right) + {\sin ^6}\left( z \right)\) .
Hint : Don’t get too locked into problems only requiring a single use of the Chain Rule. Sometimes separate terms will require different applications of the Chain Rule, or maybe only one of the terms will require the Chain Rule.
For this problem each term will require a separate application of the Chain Rule and don’t forget that,
\[{\sin ^6}\left( z \right) = {\left[ {\sin \left( z \right)} \right]^6}\]So, in the first term the outside function is the sine function, while the sine function is the inside function in the second term. The derivative is then,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{h'\left( z \right) = 6{z^5}\cos \left( {{z^6}} \right) + 6{{\sin }^5}\left( z \right)\cos \left( z \right)}}\]