3.9-bo'lim : Zanjir qoidasi
26. \(h\left( z \right) = {\tan ^4}\left( {{z^2} + 1} \right)\) ni farqlang.
Maslahat: Ba'zan biz hosilani olishni tugatishimizdan oldin zanjir qoidasini bir necha marta bajarish kerak bo'ladi.
Yechim
Ushbu muammo zanjir qoidasidan bir necha marta foydalanishni talab qiladi va shuning uchun biz har bir foydalanishni aniq qilish uchun hosila jarayoniga qadam qo'yamiz. Shuni ham esda tutingki,
\[{\tan ^4}\left( x \right) = {\left[ {\tan \left( x \right)} \right]^4}\]Mana lotinning birinchi bosqichi va biz bu bosqichda zanjir qoidasidan foydalanishimiz kerak.
\[h'\left( z \right) = 4{\tan ^3}\left( {{z^2} + 1} \right)\frac{d}{{dz}}\left[ {\tan \left( {{z^2} + 1} \right)} \right]\]2-qadam
Ko'rib turganimizdek, oldingi bosqichdan hosila ham zanjir qoidasini talab qiladi.
Demak, hosila,
\[h'\left( z \right) = 4{\tan ^3}\left( {{z^2} + 1} \right){\sec ^2}\left( {{z^2} + 1} \right)\left( {2z} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{8z{{\tan }^3}\left( {{z^2} + 1} \right){{\sec }^2}\left( {{z^2} + 1} \right)}}\]