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.
Show Solution
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,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{h'\left( z \right) = 6{z^5}\cos \left( {{z^6}} \right) + 6{{\sin }^5}\left( z \right)\cos \left( z \right)}}\]