本次共计算 1 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数lg(3{x}^{2} + {sin(x)}^{5}) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = lg(3x^{2} + sin^{5}(x))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( lg(3x^{2} + sin^{5}(x))\right)}{dx}\\=&\frac{(3*2x + 5sin^{4}(x)cos(x))}{ln{10}(3x^{2} + sin^{5}(x))}\\=&\frac{6x}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{5sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))ln{10}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{6x}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{5sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))ln{10}}\right)}{dx}\\=&\frac{6(\frac{-(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{2}})x}{ln{10}} + \frac{6}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{6x*-0}{(3x^{2} + sin^{5}(x))ln^{2}{10}} + \frac{5(\frac{-(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{2}})sin^{4}(x)cos(x)}{ln{10}} + \frac{5*-0sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))ln^{2}{10}} + \frac{5*4sin^{3}(x)cos(x)cos(x)}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{5sin^{4}(x)*-sin(x)}{(3x^{2} + sin^{5}(x))ln{10}}\\=&\frac{-60xsin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{36x^{2}}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{20sin^{3}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))ln{10}} - \frac{25sin^{8}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{5sin^{5}(x)}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{6}{(3x^{2} + sin^{5}(x))ln{10}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-60xsin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{36x^{2}}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{20sin^{3}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))ln{10}} - \frac{25sin^{8}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{5sin^{5}(x)}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{6}{(3x^{2} + sin^{5}(x))ln{10}}\right)}{dx}\\=&\frac{-60(\frac{-2(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{3}})xsin^{4}(x)cos(x)}{ln{10}} - \frac{60sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{60x*-0sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln^{2}{10}} - \frac{60x*4sin^{3}(x)cos(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{60xsin^{4}(x)*-sin(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{36(\frac{-2(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{3}})x^{2}}{ln{10}} - \frac{36*2x}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{36x^{2}*-0}{(3x^{2} + sin^{5}(x))^{2}ln^{2}{10}} + \frac{20(\frac{-(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{2}})sin^{3}(x)cos^{2}(x)}{ln{10}} + \frac{20*-0sin^{3}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))ln^{2}{10}} + \frac{20*3sin^{2}(x)cos(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{20sin^{3}(x)*-2cos(x)sin(x)}{(3x^{2} + sin^{5}(x))ln{10}} - \frac{25(\frac{-2(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{3}})sin^{8}(x)cos^{2}(x)}{ln{10}} - \frac{25*-0sin^{8}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln^{2}{10}} - \frac{25*8sin^{7}(x)cos(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{25sin^{8}(x)*-2cos(x)sin(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{5(\frac{-(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{2}})sin^{5}(x)}{ln{10}} - \frac{5*-0sin^{5}(x)}{(3x^{2} + sin^{5}(x))ln^{2}{10}} - \frac{5*5sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{6(\frac{-(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{2}})}{ln{10}} + \frac{6*-0}{(3x^{2} + sin^{5}(x))ln^{2}{10}}\\=&\frac{1080x^{2}sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{900xsin^{8}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} - \frac{90sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{360xsin^{3}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{90xsin^{5}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{432x^{3}}{(3x^{2} + sin^{5}(x))^{3}ln{10}} - \frac{108x}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{300sin^{7}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{60sin^{2}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))ln{10}} - \frac{65sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{250sin^{12}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{75sin^{9}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{1080x^{2}sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{900xsin^{8}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} - \frac{90sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{360xsin^{3}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{90xsin^{5}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{432x^{3}}{(3x^{2} + sin^{5}(x))^{3}ln{10}} - \frac{108x}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{300sin^{7}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{60sin^{2}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))ln{10}} - \frac{65sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{250sin^{12}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{75sin^{9}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}}\right)}{dx}\\=&\frac{1080(\frac{-3(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{4}})x^{2}sin^{4}(x)cos(x)}{ln{10}} + \frac{1080*2xsin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{1080x^{2}*-0sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{3}ln^{2}{10}} + \frac{1080x^{2}*4sin^{3}(x)cos(x)cos(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{1080x^{2}sin^{4}(x)*-sin(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{900(\frac{-3(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{4}})xsin^{8}(x)cos^{2}(x)}{ln{10}} + \frac{900sin^{8}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{900x*-0sin^{8}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{3}ln^{2}{10}} + \frac{900x*8sin^{7}(x)cos(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{900xsin^{8}(x)*-2cos(x)sin(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} - \frac{90(\frac{-2(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{3}})sin^{4}(x)cos(x)}{ln{10}} - \frac{90*-0sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln^{2}{10}} - \frac{90*4sin^{3}(x)cos(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{90sin^{4}(x)*-sin(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{360(\frac{-2(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{3}})xsin^{3}(x)cos^{2}(x)}{ln{10}} - \frac{360sin^{3}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{360x*-0sin^{3}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln^{2}{10}} - \frac{360x*3sin^{2}(x)cos(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{360xsin^{3}(x)*-2cos(x)sin(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{90(\frac{-2(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{3}})xsin^{5}(x)}{ln{10}} + \frac{90sin^{5}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{90x*-0sin^{5}(x)}{(3x^{2} + sin^{5}(x))^{2}ln^{2}{10}} + \frac{90x*5sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{432(\frac{-3(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{4}})x^{3}}{ln{10}} + \frac{432*3x^{2}}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{432x^{3}*-0}{(3x^{2} + sin^{5}(x))^{3}ln^{2}{10}} - \frac{108(\frac{-2(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{3}})x}{ln{10}} - \frac{108}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{108x*-0}{(3x^{2} + sin^{5}(x))^{2}ln^{2}{10}} - \frac{300(\frac{-2(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{3}})sin^{7}(x)cos^{3}(x)}{ln{10}} - \frac{300*-0sin^{7}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{2}ln^{2}{10}} - \frac{300*7sin^{6}(x)cos(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{300sin^{7}(x)*-3cos^{2}(x)sin(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{60(\frac{-(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{2}})sin^{2}(x)cos^{3}(x)}{ln{10}} + \frac{60*-0sin^{2}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))ln^{2}{10}} + \frac{60*2sin(x)cos(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{60sin^{2}(x)*-3cos^{2}(x)sin(x)}{(3x^{2} + sin^{5}(x))ln{10}} - \frac{65(\frac{-(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{2}})sin^{4}(x)cos(x)}{ln{10}} - \frac{65*-0sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))ln^{2}{10}} - \frac{65*4sin^{3}(x)cos(x)cos(x)}{(3x^{2} + sin^{5}(x))ln{10}} - \frac{65sin^{4}(x)*-sin(x)}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{250(\frac{-3(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{4}})sin^{12}(x)cos^{3}(x)}{ln{10}} + \frac{250*-0sin^{12}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{3}ln^{2}{10}} + \frac{250*12sin^{11}(x)cos(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{250sin^{12}(x)*-3cos^{2}(x)sin(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{75(\frac{-2(3*2x + 5sin^{4}(x)cos(x))}{(3x^{2} + sin^{5}(x))^{3}})sin^{9}(x)cos(x)}{ln{10}} + \frac{75*-0sin^{9}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln^{2}{10}} + \frac{75*9sin^{8}(x)cos(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{75sin^{9}(x)*-sin(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}}\\=&\frac{-25920x^{3}sin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{4}ln{10}} - \frac{32400x^{2}sin^{8}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{4}ln{10}} + \frac{4320xsin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{8640x^{2}sin^{3}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} - \frac{3600xsin^{9}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} - \frac{18000xsin^{12}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{4}ln{10}} + \frac{1800sin^{8}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{14400xsin^{7}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} - \frac{1440xsin^{2}(x)cos^{3}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{1560xsin^{4}(x)cos(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{720sin^{3}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{2400sin^{6}(x)cos^{4}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{1900sin^{8}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{2160x^{2}sin^{5}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{6000sin^{11}(x)cos^{4}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} - \frac{7776x^{4}}{(3x^{2} + sin^{5}(x))^{4}ln{10}} + \frac{2592x^{2}}{(3x^{2} + sin^{5}(x))^{3}ln{10}} + \frac{120sin(x)cos^{4}(x)}{(3x^{2} + sin^{5}(x))ln{10}} - \frac{440sin^{3}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))ln{10}} - \frac{1500sin^{13}(x)cos^{2}(x)}{(3x^{2} + sin^{5}(x))^{3}ln{10}} - \frac{3750sin^{16}(x)cos^{4}(x)}{(3x^{2} + sin^{5}(x))^{4}ln{10}} - \frac{75sin^{10}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} + \frac{65sin^{5}(x)}{(3x^{2} + sin^{5}(x))ln{10}} + \frac{180sin^{5}(x)}{(3x^{2} + sin^{5}(x))^{2}ln{10}} - \frac{108}{(3x^{2} + sin^{5}(x))^{2}ln{10}}\\ \end{split}\end{equation} \]

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