There are 1 questions in this calculation: for each question, the 15 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 15th\ derivative\ of\ function\ {(sin(x))}^{n}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {sin(x)}^{n}\\\\ &\color{blue}{The\ 15th\ derivative\ of\ function:} \\=&\frac{n^{15}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} - \frac{105n^{14}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} - \frac{105n^{14}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} + \frac{5005n^{13}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} + \frac{9100n^{13}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} + \frac{4095n^{13}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} - \frac{143325n^{12}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} - \frac{356265n^{12}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} - \frac{288015n^{12}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} - \frac{75075n^{12}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} + \frac{2749747n^{11}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} + \frac{8328320n^{11}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} + \frac{9107098n^{11}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} + \frac{4204200n^{11}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} + \frac{675675n^{11}{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} - \frac{37312275n^{10}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} - \frac{129444315n^{10}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} - \frac{170860690n^{10}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} - \frac{105395290n^{10}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} - \frac{29504475n^{10}{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} - \frac{2837835n^{10}{sin(x)}^{n}cos^{5}(x)}{sin^{5}(x)} + \frac{368411615n^{9}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} + \frac{1409307900n^{9}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} + \frac{2113406295n^{9}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} + \frac{1557464480n^{9}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} + \frac{574819245n^{9}{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} + \frac{94594500n^{9}{sin(x)}^{n}cos^{5}(x)}{sin^{5}(x)} + \frac{4729725n^{9}{sin(x)}^{n}cos^{3}(x)}{sin^{3}(x)} - \frac{2681453775n^{8}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} - \frac{11027211195n^{8}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} - \frac{18106183095n^{8}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} - \frac{15029197755n^{8}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} - \frac{6556023045n^{8}{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} - \frac{1403467065n^{8}{sin(x)}^{n}cos^{5}(x)}{sin^{5}(x)} - \frac{118243125n^{8}{sin(x)}^{n}cos^{3}(x)}{sin^{3}(x)} - \frac{2027025n^{8}{sin(x)}^{n}cos(x)}{sin(x)} + \frac{14409322928n^{7}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} + \frac{62578475960n^{7}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} + \frac{109799193504n^{7}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} + \frac{99080861880n^{7}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} + \frac{48268107888n^{7}{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} + \frac{12091159080n^{7}{sin(x)}^{n}cos^{5}(x)}{sin^{5}(x)} + \frac{1311710400n^{7}{sin(x)}^{n}cos^{3}(x)}{sin^{3}(x)} + \frac{37837800n^{7}{sin(x)}^{n}cos(x)}{sin(x)} - \frac{256530554280n^{6}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} - \frac{472919286840n^{6}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} - \frac{56663366760n^{6}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} - \frac{453350297400n^{6}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} - \frac{238476085848n^{6}{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} - \frac{66215357208n^{6}{sin(x)}^{n}cos^{5}(x)}{sin^{5}(x)} - \frac{8347739400n^{6}{sin(x)}^{n}cos^{3}(x)}{sin^{3}(x)} - \frac{310269960n^{6}{sin(x)}^{n}cos(x)}{sin(x)} + \frac{1428504437360n^{5}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} + \frac{1431846736320n^{5}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} + \frac{795830315280n^{5}{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} + \frac{746529784000n^{5}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} + \frac{237245211840n^{5}{sin(x)}^{n}cos^{5}(x)}{sin^{5}(x)} + \frac{159721605680n^{5}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} + \frac{32992399440n^{5}{sin(x)}^{n}cos^{3}(x)}{sin^{3}(x)} + \frac{1427025600n^{5}{sin(x)}^{n}cos(x)}{sin(x)} - \frac{3038180665520n^{4}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} - \frac{1757975178960n^{4}{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} - \frac{551277002640n^{4}{sin(x)}^{n}cos^{5}(x)}{sin^{5}(x)} - \frac{82016210640n^{4}{sin(x)}^{n}cos^{3}(x)}{sin^{3}(x)} - \frac{2931962873840n^{4}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} - \frac{3918554640n^{4}{sin(x)}^{n}cos(x)}{sin(x)} - \frac{1489567301040n^{4}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} + \frac{2436549827712n^{3}{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} + \frac{792552808320n^{3}{sin(x)}^{n}cos^{5}(x)}{sin^{5}(x)} + \frac{123675031936n^{3}{sin(x)}^{n}cos^{3}(x)}{sin^{3}(x)} - \frac{310989260400n^{4}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} + \frac{6327135360n^{3}{sin(x)}^{n}cos(x)}{sin(x)} + \frac{4087987995520n^{3}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} + \frac{3848281976448n^{3}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} - \frac{630895679232n^{2}{sin(x)}^{n}cos^{5}(x)}{sin^{5}(x)} - \frac{101844649728n^{2}{sin(x)}^{n}cos^{3}(x)}{sin^{3}(x)} - \frac{5464904448n^{2}{sin(x)}^{n}cos(x)}{sin(x)} - \frac{1888988504832n^{2}{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} + \frac{1913795694720n^{3}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} + \frac{34342971392n{sin(x)}^{n}cos^{3}(x)}{sin^{3}(x)} + \frac{1903757312n{sin(x)}^{n}cos(x)}{sin(x)} - \frac{3101371004160n^{2}{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} + \frac{207810570240n{sin(x)}^{n}cos^{5}(x)}{sin^{5}(x)} + \frac{610801551360n{sin(x)}^{n}cos^{7}(x)}{sin^{7}(x)} + \frac{987559372800n{sin(x)}^{n}cos^{9}(x)}{sin^{9}(x)} + \frac{392156797824n^{3}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} - \frac{2866390974720n^{2}{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} + \frac{900842342400n{sin(x)}^{n}cos^{11}(x)}{sin^{11}(x)} - \frac{1402958188800n^{2}{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} + \frac{435891456000n{sin(x)}^{n}cos^{13}(x)}{sin^{13}(x)} - \frac{283465647360n^{2}{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)} + \frac{87178291200n{sin(x)}^{n}cos^{15}(x)}{sin^{15}(x)}\\ \end{split}\end{equation} \]

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