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
3. Differentiate \(y = \sqrt[3]{{1 - 8z}}\) .
Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule.
For this problem, after converting the root to a fractional exponent, the outside function is (hopefully) clearly the exponent of \(\frac{1}{3}\) while the inside function is the polynomial that is being raised to the power (or the polynomial inside the root – depending upon how you want to think about it). The derivative is then,
\[y = {\left( {1 - 8z} \right)^{\frac{1}{3}}}\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\,\,\,\,\,\,\,\,\,\,\frac{{dy}}{{dz}} = \frac{1}{3}{\left( {1 - 8z} \right)^{ - \,\,\frac{2}{3}}}\left( { - 8} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{8}{3}{{\left( {1 - 8z} \right)}^{ - \,\,\frac{2}{3}}}}}\]