10.15-bo'lim : Quvvat seriyalari va funktsiyalari
4. Quyidagi funksiyaning hosilasi uchun daraja qatori tasvirini keltiring.
\[g\left( x \right) = \frac{{5x}}{{1 - 3{x^5}}}\]
Maslahat: Biz funktsiyani farqlashimiz va keyin juda ko'p ish kabi ko'rinadigan quvvat seriyasining tasvirini topishga harakat qilishimiz mumkin edi. Quvvat seriyalarini qanday ajratishni bilganimiz yaxshi.
Yechim
Avval shuni ta'kidlaymizki, biz ushbu funktsiya uchun quvvat seriyasining tasvirini tezda topishimiz mumkin. Mana o'sha ish.
\[g\left( x \right) = 5x\frac{1}{{1 - 3{x^5}}} = 5x\sum\limits_{n = 0}^\infty {{{\left( {3{x^5}} \right)}^n}} = \sum\limits_{n = 0}^\infty {5x\,\left( {{3^n}} \right){x^{5n}}} = \sum\limits_{n = 0}^\infty {5\left( {{3^n}} \right){x^{5n + 1}}} \]2-qadam
Endi biz darajali qatorlarni qanday ajratishni bilamiz va biz bilamizki, funktsiyaning darajali qator ko'rinishining hosilasi funktsiya hosilasining darajali seriyali ko'rinishidir.
Shuning uchun,
\[g'\left( x \right) = \frac{d}{{dx}}\left[ {\sum\limits_{n = 0}^\infty {5\left( {{3^n}} \right){x^{5n + 1}}} } \right] = \require{bbox} \bbox[2pt,border:1px solid black]{{\sum\limits_{n = 0}^\infty {5\left( {5n + 1} \right)\left( {{3^n}} \right){x^{5n}}} }}\]Esda tutingki, kuch seriyasini farqlash uchun biz faqat \(x\) ga nisbatan quvvat seriyasining atamasini farqlashimiz kerak.