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。
\[ \begin{equation}\begin{split}【1/1】求函数{(3x)}^{sin(5x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = (3x)^{sin(5x)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( (3x)^{sin(5x)}\right)}{dx}\\=&((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))\\=&5(3x)^{sin(5x)}ln(3x)cos(5x) + \frac{(3x)^{sin(5x)}sin(5x)}{x}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( 5(3x)^{sin(5x)}ln(3x)cos(5x) + \frac{(3x)^{sin(5x)}sin(5x)}{x}\right)}{dx}\\=&5((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln(3x)cos(5x) + \frac{5(3x)^{sin(5x)}*3cos(5x)}{(3x)} + 5(3x)^{sin(5x)}ln(3x)*-sin(5x)*5 + \frac{-(3x)^{sin(5x)}sin(5x)}{x^{2}} + \frac{((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))sin(5x)}{x} + \frac{(3x)^{sin(5x)}cos(5x)*5}{x}\\=&25(3x)^{sin(5x)}ln^{2}(3x)cos^{2}(5x) + \frac{10(3x)^{sin(5x)}ln(3x)sin(5x)cos(5x)}{x} + \frac{10(3x)^{sin(5x)}cos(5x)}{x} - 25(3x)^{sin(5x)}ln(3x)sin(5x) - \frac{(3x)^{sin(5x)}sin(5x)}{x^{2}} + \frac{(3x)^{sin(5x)}sin^{2}(5x)}{x^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 25(3x)^{sin(5x)}ln^{2}(3x)cos^{2}(5x) + \frac{10(3x)^{sin(5x)}ln(3x)sin(5x)cos(5x)}{x} + \frac{10(3x)^{sin(5x)}cos(5x)}{x} - 25(3x)^{sin(5x)}ln(3x)sin(5x) - \frac{(3x)^{sin(5x)}sin(5x)}{x^{2}} + \frac{(3x)^{sin(5x)}sin^{2}(5x)}{x^{2}}\right)}{dx}\\=&25((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln^{2}(3x)cos^{2}(5x) + \frac{25(3x)^{sin(5x)}*2ln(3x)*3cos^{2}(5x)}{(3x)} + 25(3x)^{sin(5x)}ln^{2}(3x)*-2cos(5x)sin(5x)*5 + \frac{10*-(3x)^{sin(5x)}ln(3x)sin(5x)cos(5x)}{x^{2}} + \frac{10((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln(3x)sin(5x)cos(5x)}{x} + \frac{10(3x)^{sin(5x)}*3sin(5x)cos(5x)}{x(3x)} + \frac{10(3x)^{sin(5x)}ln(3x)cos(5x)*5cos(5x)}{x} + \frac{10(3x)^{sin(5x)}ln(3x)sin(5x)*-sin(5x)*5}{x} + \frac{10*-(3x)^{sin(5x)}cos(5x)}{x^{2}} + \frac{10((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))cos(5x)}{x} + \frac{10(3x)^{sin(5x)}*-sin(5x)*5}{x} - 25((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln(3x)sin(5x) - \frac{25(3x)^{sin(5x)}*3sin(5x)}{(3x)} - 25(3x)^{sin(5x)}ln(3x)cos(5x)*5 - \frac{-2(3x)^{sin(5x)}sin(5x)}{x^{3}} - \frac{((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))sin(5x)}{x^{2}} - \frac{(3x)^{sin(5x)}cos(5x)*5}{x^{2}} + \frac{-2(3x)^{sin(5x)}sin^{2}(5x)}{x^{3}} + \frac{((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))sin^{2}(5x)}{x^{2}} + \frac{(3x)^{sin(5x)}*2sin(5x)cos(5x)*5}{x^{2}}\\=&125(3x)^{sin(5x)}ln^{3}(3x)cos^{3}(5x) + \frac{75(3x)^{sin(5x)}ln^{2}(3x)sin(5x)cos^{2}(5x)}{x} + \frac{150(3x)^{sin(5x)}ln(3x)cos^{2}(5x)}{x} - 375(3x)^{sin(5x)}ln^{2}(3x)sin(5x)cos(5x) - \frac{15(3x)^{sin(5x)}ln(3x)sin(5x)cos(5x)}{x^{2}} + \frac{15(3x)^{sin(5x)}ln(3x)sin^{2}(5x)cos(5x)}{x^{2}} + \frac{30(3x)^{sin(5x)}sin(5x)cos(5x)}{x^{2}} - \frac{75(3x)^{sin(5x)}ln(3x)sin^{2}(5x)}{x} - \frac{15(3x)^{sin(5x)}cos(5x)}{x^{2}} - \frac{75(3x)^{sin(5x)}sin(5x)}{x} - 125(3x)^{sin(5x)}ln(3x)cos(5x) + \frac{2(3x)^{sin(5x)}sin(5x)}{x^{3}} - \frac{3(3x)^{sin(5x)}sin^{2}(5x)}{x^{3}} + \frac{(3x)^{sin(5x)}sin^{3}(5x)}{x^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 125(3x)^{sin(5x)}ln^{3}(3x)cos^{3}(5x) + \frac{75(3x)^{sin(5x)}ln^{2}(3x)sin(5x)cos^{2}(5x)}{x} + \frac{150(3x)^{sin(5x)}ln(3x)cos^{2}(5x)}{x} - 375(3x)^{sin(5x)}ln^{2}(3x)sin(5x)cos(5x) - \frac{15(3x)^{sin(5x)}ln(3x)sin(5x)cos(5x)}{x^{2}} + \frac{15(3x)^{sin(5x)}ln(3x)sin^{2}(5x)cos(5x)}{x^{2}} + \frac{30(3x)^{sin(5x)}sin(5x)cos(5x)}{x^{2}} - \frac{75(3x)^{sin(5x)}ln(3x)sin^{2}(5x)}{x} - \frac{15(3x)^{sin(5x)}cos(5x)}{x^{2}} - \frac{75(3x)^{sin(5x)}sin(5x)}{x} - 125(3x)^{sin(5x)}ln(3x)cos(5x) + \frac{2(3x)^{sin(5x)}sin(5x)}{x^{3}} - \frac{3(3x)^{sin(5x)}sin^{2}(5x)}{x^{3}} + \frac{(3x)^{sin(5x)}sin^{3}(5x)}{x^{3}}\right)}{dx}\\=&125((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln^{3}(3x)cos^{3}(5x) + \frac{125(3x)^{sin(5x)}*3ln^{2}(3x)*3cos^{3}(5x)}{(3x)} + 125(3x)^{sin(5x)}ln^{3}(3x)*-3cos^{2}(5x)sin(5x)*5 + \frac{75*-(3x)^{sin(5x)}ln^{2}(3x)sin(5x)cos^{2}(5x)}{x^{2}} + \frac{75((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln^{2}(3x)sin(5x)cos^{2}(5x)}{x} + \frac{75(3x)^{sin(5x)}*2ln(3x)*3sin(5x)cos^{2}(5x)}{x(3x)} + \frac{75(3x)^{sin(5x)}ln^{2}(3x)cos(5x)*5cos^{2}(5x)}{x} + \frac{75(3x)^{sin(5x)}ln^{2}(3x)sin(5x)*-2cos(5x)sin(5x)*5}{x} + \frac{150*-(3x)^{sin(5x)}ln(3x)cos^{2}(5x)}{x^{2}} + \frac{150((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln(3x)cos^{2}(5x)}{x} + \frac{150(3x)^{sin(5x)}*3cos^{2}(5x)}{x(3x)} + \frac{150(3x)^{sin(5x)}ln(3x)*-2cos(5x)sin(5x)*5}{x} - 375((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln^{2}(3x)sin(5x)cos(5x) - \frac{375(3x)^{sin(5x)}*2ln(3x)*3sin(5x)cos(5x)}{(3x)} - 375(3x)^{sin(5x)}ln^{2}(3x)cos(5x)*5cos(5x) - 375(3x)^{sin(5x)}ln^{2}(3x)sin(5x)*-sin(5x)*5 - \frac{15*-2(3x)^{sin(5x)}ln(3x)sin(5x)cos(5x)}{x^{3}} - \frac{15((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln(3x)sin(5x)cos(5x)}{x^{2}} - \frac{15(3x)^{sin(5x)}*3sin(5x)cos(5x)}{x^{2}(3x)} - \frac{15(3x)^{sin(5x)}ln(3x)cos(5x)*5cos(5x)}{x^{2}} - \frac{15(3x)^{sin(5x)}ln(3x)sin(5x)*-sin(5x)*5}{x^{2}} + \frac{15*-2(3x)^{sin(5x)}ln(3x)sin^{2}(5x)cos(5x)}{x^{3}} + \frac{15((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln(3x)sin^{2}(5x)cos(5x)}{x^{2}} + \frac{15(3x)^{sin(5x)}*3sin^{2}(5x)cos(5x)}{x^{2}(3x)} + \frac{15(3x)^{sin(5x)}ln(3x)*2sin(5x)cos(5x)*5cos(5x)}{x^{2}} + \frac{15(3x)^{sin(5x)}ln(3x)sin^{2}(5x)*-sin(5x)*5}{x^{2}} + \frac{30*-2(3x)^{sin(5x)}sin(5x)cos(5x)}{x^{3}} + \frac{30((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))sin(5x)cos(5x)}{x^{2}} + \frac{30(3x)^{sin(5x)}cos(5x)*5cos(5x)}{x^{2}} + \frac{30(3x)^{sin(5x)}sin(5x)*-sin(5x)*5}{x^{2}} - \frac{75*-(3x)^{sin(5x)}ln(3x)sin^{2}(5x)}{x^{2}} - \frac{75((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln(3x)sin^{2}(5x)}{x} - \frac{75(3x)^{sin(5x)}*3sin^{2}(5x)}{x(3x)} - \frac{75(3x)^{sin(5x)}ln(3x)*2sin(5x)cos(5x)*5}{x} - \frac{15*-2(3x)^{sin(5x)}cos(5x)}{x^{3}} - \frac{15((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))cos(5x)}{x^{2}} - \frac{15(3x)^{sin(5x)}*-sin(5x)*5}{x^{2}} - \frac{75*-(3x)^{sin(5x)}sin(5x)}{x^{2}} - \frac{75((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))sin(5x)}{x} - \frac{75(3x)^{sin(5x)}cos(5x)*5}{x} - 125((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))ln(3x)cos(5x) - \frac{125(3x)^{sin(5x)}*3cos(5x)}{(3x)} - 125(3x)^{sin(5x)}ln(3x)*-sin(5x)*5 + \frac{2*-3(3x)^{sin(5x)}sin(5x)}{x^{4}} + \frac{2((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))sin(5x)}{x^{3}} + \frac{2(3x)^{sin(5x)}cos(5x)*5}{x^{3}} - \frac{3*-3(3x)^{sin(5x)}sin^{2}(5x)}{x^{4}} - \frac{3((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))sin^{2}(5x)}{x^{3}} - \frac{3(3x)^{sin(5x)}*2sin(5x)cos(5x)*5}{x^{3}} + \frac{-3(3x)^{sin(5x)}sin^{3}(5x)}{x^{4}} + \frac{((3x)^{sin(5x)}((cos(5x)*5)ln(3x) + \frac{(sin(5x))(3)}{(3x)}))sin^{3}(5x)}{x^{3}} + \frac{(3x)^{sin(5x)}*3sin^{2}(5x)cos(5x)*5}{x^{3}}\\=&625(3x)^{sin(5x)}ln^{4}(3x)cos^{4}(5x) + \frac{600(3x)^{sin(5x)}ln(3x)sin(5x)cos^{2}(5x)}{x^{2}} + \frac{1500(3x)^{sin(5x)}ln^{2}(3x)cos^{3}(5x)}{x} - 3750(3x)^{sin(5x)}ln^{3}(3x)sin(5x)cos^{2}(5x) - \frac{150(3x)^{sin(5x)}ln^{2}(3x)sin(5x)cos^{2}(5x)}{x^{2}} + \frac{500(3x)^{sin(5x)}ln^{3}(3x)sin(5x)cos^{3}(5x)}{x} + \frac{150(3x)^{sin(5x)}ln^{2}(3x)sin^{2}(5x)cos^{2}(5x)}{x^{2}} - \frac{1500(3x)^{sin(5x)}ln^{2}(3x)sin^{2}(5x)cos(5x)}{x} - \frac{300(3x)^{sin(5x)}ln(3x)cos^{2}(5x)}{x^{2}} + \frac{300(3x)^{sin(5x)}cos^{2}(5x)}{x^{2}} - \frac{3500(3x)^{sin(5x)}ln(3x)sin(5x)cos(5x)}{x} - 2500(3x)^{sin(5x)}ln^{2}(3x)cos^{2}(5x) + 1875(3x)^{sin(5x)}ln^{2}(3x)sin^{2}(5x) + \frac{40(3x)^{sin(5x)}ln(3x)sin(5x)cos(5x)}{x^{3}} - \frac{60(3x)^{sin(5x)}ln(3x)sin^{2}(5x)cos(5x)}{x^{3}} - \frac{120(3x)^{sin(5x)}sin(5x)cos(5x)}{x^{3}} + \frac{20(3x)^{sin(5x)}ln(3x)sin^{3}(5x)cos(5x)}{x^{3}} + \frac{150(3x)^{sin(5x)}ln(3x)sin^{2}(5x)}{x^{2}} + \frac{60(3x)^{sin(5x)}sin^{2}(5x)cos(5x)}{x^{3}} - \frac{150(3x)^{sin(5x)}ln(3x)sin^{3}(5x)}{x^{2}} - \frac{300(3x)^{sin(5x)}sin^{2}(5x)}{x^{2}} + \frac{40(3x)^{sin(5x)}cos(5x)}{x^{3}} + \frac{150(3x)^{sin(5x)}sin(5x)}{x^{2}} - \frac{500(3x)^{sin(5x)}cos(5x)}{x} + 625(3x)^{sin(5x)}ln(3x)sin(5x) - \frac{6(3x)^{sin(5x)}sin(5x)}{x^{4}} + \frac{11(3x)^{sin(5x)}sin^{2}(5x)}{x^{4}} - \frac{6(3x)^{sin(5x)}sin^{3}(5x)}{x^{4}} + \frac{(3x)^{sin(5x)}sin^{4}(5x)}{x^{4}}\\ \end{split}\end{equation} \]

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