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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.
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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,

$y = {\left( {1 - 8z} \right)^{\frac{1}{3}}}\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\,\,\,\,\,\,\,\,\,\,\frac{{dy}}{{dz}} = \frac{1}{3}{\left( {1 - 8z} \right)^{ - \,\,\frac{2}{3}}}\left( { - 8} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{8}{3}{{\left( {1 - 8z} \right)}^{ - \,\,\frac{2}{3}}}}}$