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。
\[ \begin{equation}\begin{split}【1/1】求函数{(sin(x))}^{x} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = {sin(x)}^{x}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {sin(x)}^{x}\right)}{dx}\\=&({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))\\=&{sin(x)}^{x}ln(sin(x)) + \frac{x{sin(x)}^{x}cos(x)}{sin(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( {sin(x)}^{x}ln(sin(x)) + \frac{x{sin(x)}^{x}cos(x)}{sin(x)}\right)}{dx}\\=&({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))ln(sin(x)) + \frac{{sin(x)}^{x}cos(x)}{(sin(x))} + \frac{{sin(x)}^{x}cos(x)}{sin(x)} + \frac{x({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos(x)}{sin(x)} + \frac{x{sin(x)}^{x}*-cos(x)cos(x)}{sin^{2}(x)} + \frac{x{sin(x)}^{x}*-sin(x)}{sin(x)}\\=&{sin(x)}^{x}ln^{2}(sin(x)) + \frac{2x{sin(x)}^{x}ln(sin(x))cos(x)}{sin(x)} + \frac{2{sin(x)}^{x}cos(x)}{sin(x)} + \frac{x^{2}{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - \frac{x{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - x{sin(x)}^{x}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( {sin(x)}^{x}ln^{2}(sin(x)) + \frac{2x{sin(x)}^{x}ln(sin(x))cos(x)}{sin(x)} + \frac{2{sin(x)}^{x}cos(x)}{sin(x)} + \frac{x^{2}{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - \frac{x{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - x{sin(x)}^{x}\right)}{dx}\\=&({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))ln^{2}(sin(x)) + \frac{{sin(x)}^{x}*2ln(sin(x))cos(x)}{(sin(x))} + \frac{2{sin(x)}^{x}ln(sin(x))cos(x)}{sin(x)} + \frac{2x({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))ln(sin(x))cos(x)}{sin(x)} + \frac{2x{sin(x)}^{x}cos(x)cos(x)}{(sin(x))sin(x)} + \frac{2x{sin(x)}^{x}ln(sin(x))*-cos(x)cos(x)}{sin^{2}(x)} + \frac{2x{sin(x)}^{x}ln(sin(x))*-sin(x)}{sin(x)} + \frac{2({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos(x)}{sin(x)} + \frac{2{sin(x)}^{x}*-cos(x)cos(x)}{sin^{2}(x)} + \frac{2{sin(x)}^{x}*-sin(x)}{sin(x)} + \frac{2x{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} + \frac{x^{2}({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos^{2}(x)}{sin^{2}(x)} + \frac{x^{2}{sin(x)}^{x}*-2cos(x)cos^{2}(x)}{sin^{3}(x)} + \frac{x^{2}{sin(x)}^{x}*-2cos(x)sin(x)}{sin^{2}(x)} - \frac{{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - \frac{x({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos^{2}(x)}{sin^{2}(x)} - \frac{x{sin(x)}^{x}*-2cos(x)cos^{2}(x)}{sin^{3}(x)} - \frac{x{sin(x)}^{x}*-2cos(x)sin(x)}{sin^{2}(x)} - {sin(x)}^{x} - x({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))\\=&\frac{6{sin(x)}^{x}ln(sin(x))cos(x)}{sin(x)} + \frac{3x{sin(x)}^{x}ln^{2}(sin(x))cos(x)}{sin(x)} + {sin(x)}^{x}ln^{3}(sin(x)) + \frac{3x^{2}{sin(x)}^{x}ln(sin(x))cos^{2}(x)}{sin^{2}(x)} + \frac{6x{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - \frac{3x{sin(x)}^{x}ln(sin(x))cos^{2}(x)}{sin^{2}(x)} - 3x{sin(x)}^{x}ln(sin(x)) - \frac{3{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - 3{sin(x)}^{x} + \frac{x^{3}{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} - \frac{3x^{2}{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} - \frac{3x^{2}{sin(x)}^{x}cos(x)}{sin(x)} + \frac{2x{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} + \frac{2x{sin(x)}^{x}cos(x)}{sin(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{6{sin(x)}^{x}ln(sin(x))cos(x)}{sin(x)} + \frac{3x{sin(x)}^{x}ln^{2}(sin(x))cos(x)}{sin(x)} + {sin(x)}^{x}ln^{3}(sin(x)) + \frac{3x^{2}{sin(x)}^{x}ln(sin(x))cos^{2}(x)}{sin^{2}(x)} + \frac{6x{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - \frac{3x{sin(x)}^{x}ln(sin(x))cos^{2}(x)}{sin^{2}(x)} - 3x{sin(x)}^{x}ln(sin(x)) - \frac{3{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - 3{sin(x)}^{x} + \frac{x^{3}{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} - \frac{3x^{2}{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} - \frac{3x^{2}{sin(x)}^{x}cos(x)}{sin(x)} + \frac{2x{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} + \frac{2x{sin(x)}^{x}cos(x)}{sin(x)}\right)}{dx}\\=&\frac{6({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))ln(sin(x))cos(x)}{sin(x)} + \frac{6{sin(x)}^{x}cos(x)cos(x)}{(sin(x))sin(x)} + \frac{6{sin(x)}^{x}ln(sin(x))*-cos(x)cos(x)}{sin^{2}(x)} + \frac{6{sin(x)}^{x}ln(sin(x))*-sin(x)}{sin(x)} + \frac{3{sin(x)}^{x}ln^{2}(sin(x))cos(x)}{sin(x)} + \frac{3x({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))ln^{2}(sin(x))cos(x)}{sin(x)} + \frac{3x{sin(x)}^{x}*2ln(sin(x))cos(x)cos(x)}{(sin(x))sin(x)} + \frac{3x{sin(x)}^{x}ln^{2}(sin(x))*-cos(x)cos(x)}{sin^{2}(x)} + \frac{3x{sin(x)}^{x}ln^{2}(sin(x))*-sin(x)}{sin(x)} + ({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))ln^{3}(sin(x)) + \frac{{sin(x)}^{x}*3ln^{2}(sin(x))cos(x)}{(sin(x))} + \frac{3*2x{sin(x)}^{x}ln(sin(x))cos^{2}(x)}{sin^{2}(x)} + \frac{3x^{2}({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))ln(sin(x))cos^{2}(x)}{sin^{2}(x)} + \frac{3x^{2}{sin(x)}^{x}cos(x)cos^{2}(x)}{(sin(x))sin^{2}(x)} + \frac{3x^{2}{sin(x)}^{x}ln(sin(x))*-2cos(x)cos^{2}(x)}{sin^{3}(x)} + \frac{3x^{2}{sin(x)}^{x}ln(sin(x))*-2cos(x)sin(x)}{sin^{2}(x)} + \frac{6{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} + \frac{6x({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos^{2}(x)}{sin^{2}(x)} + \frac{6x{sin(x)}^{x}*-2cos(x)cos^{2}(x)}{sin^{3}(x)} + \frac{6x{sin(x)}^{x}*-2cos(x)sin(x)}{sin^{2}(x)} - \frac{3{sin(x)}^{x}ln(sin(x))cos^{2}(x)}{sin^{2}(x)} - \frac{3x({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))ln(sin(x))cos^{2}(x)}{sin^{2}(x)} - \frac{3x{sin(x)}^{x}cos(x)cos^{2}(x)}{(sin(x))sin^{2}(x)} - \frac{3x{sin(x)}^{x}ln(sin(x))*-2cos(x)cos^{2}(x)}{sin^{3}(x)} - \frac{3x{sin(x)}^{x}ln(sin(x))*-2cos(x)sin(x)}{sin^{2}(x)} - 3{sin(x)}^{x}ln(sin(x)) - 3x({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))ln(sin(x)) - \frac{3x{sin(x)}^{x}cos(x)}{(sin(x))} - \frac{3({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos^{2}(x)}{sin^{2}(x)} - \frac{3{sin(x)}^{x}*-2cos(x)cos^{2}(x)}{sin^{3}(x)} - \frac{3{sin(x)}^{x}*-2cos(x)sin(x)}{sin^{2}(x)} - 3({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))})) + \frac{3x^{2}{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} + \frac{x^{3}({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos^{3}(x)}{sin^{3}(x)} + \frac{x^{3}{sin(x)}^{x}*-3cos(x)cos^{3}(x)}{sin^{4}(x)} + \frac{x^{3}{sin(x)}^{x}*-3cos^{2}(x)sin(x)}{sin^{3}(x)} - \frac{3*2x{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} - \frac{3x^{2}({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos^{3}(x)}{sin^{3}(x)} - \frac{3x^{2}{sin(x)}^{x}*-3cos(x)cos^{3}(x)}{sin^{4}(x)} - \frac{3x^{2}{sin(x)}^{x}*-3cos^{2}(x)sin(x)}{sin^{3}(x)} - \frac{3*2x{sin(x)}^{x}cos(x)}{sin(x)} - \frac{3x^{2}({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos(x)}{sin(x)} - \frac{3x^{2}{sin(x)}^{x}*-cos(x)cos(x)}{sin^{2}(x)} - \frac{3x^{2}{sin(x)}^{x}*-sin(x)}{sin(x)} + \frac{2{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} + \frac{2x({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos^{3}(x)}{sin^{3}(x)} + \frac{2x{sin(x)}^{x}*-3cos(x)cos^{3}(x)}{sin^{4}(x)} + \frac{2x{sin(x)}^{x}*-3cos^{2}(x)sin(x)}{sin^{3}(x)} + \frac{2{sin(x)}^{x}cos(x)}{sin(x)} + \frac{2x({sin(x)}^{x}((1)ln(sin(x)) + \frac{(x)(cos(x))}{(sin(x))}))cos(x)}{sin(x)} + \frac{2x{sin(x)}^{x}*-cos(x)cos(x)}{sin^{2}(x)} + \frac{2x{sin(x)}^{x}*-sin(x)}{sin(x)}\\=&\frac{12{sin(x)}^{x}ln^{2}(sin(x))cos(x)}{sin(x)} + \frac{24x{sin(x)}^{x}ln(sin(x))cos^{2}(x)}{sin^{2}(x)} + \frac{12{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - \frac{12{sin(x)}^{x}ln(sin(x))cos^{2}(x)}{sin^{2}(x)} - 12{sin(x)}^{x}ln(sin(x)) + \frac{4x{sin(x)}^{x}ln^{3}(sin(x))cos(x)}{sin(x)} + \frac{6x^{2}{sin(x)}^{x}ln^{2}(sin(x))cos^{2}(x)}{sin^{2}(x)} - \frac{6x{sin(x)}^{x}ln^{2}(sin(x))cos^{2}(x)}{sin^{2}(x)} - \frac{12x^{2}{sin(x)}^{x}ln(sin(x))cos(x)}{sin(x)} + {sin(x)}^{x}ln^{4}(sin(x)) + \frac{4x^{3}{sin(x)}^{x}ln(sin(x))cos^{3}(x)}{sin^{3}(x)} + \frac{12x^{2}{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} - \frac{12x^{2}{sin(x)}^{x}ln(sin(x))cos^{3}(x)}{sin^{3}(x)} + \frac{8x{sin(x)}^{x}ln(sin(x))cos(x)}{sin(x)} - \frac{24x{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} - \frac{24x{sin(x)}^{x}cos(x)}{sin(x)} + \frac{8x{sin(x)}^{x}ln(sin(x))cos^{3}(x)}{sin^{3}(x)} - 6x{sin(x)}^{x}ln^{2}(sin(x)) + \frac{8{sin(x)}^{x}cos^{3}(x)}{sin^{3}(x)} + \frac{8{sin(x)}^{x}cos(x)}{sin(x)} + \frac{x^{4}{sin(x)}^{x}cos^{4}(x)}{sin^{4}(x)} - \frac{6x^{3}{sin(x)}^{x}cos^{4}(x)}{sin^{4}(x)} - \frac{6x^{3}{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} + \frac{11x^{2}{sin(x)}^{x}cos^{4}(x)}{sin^{4}(x)} + \frac{14x^{2}{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - \frac{8x{sin(x)}^{x}cos^{2}(x)}{sin^{2}(x)} - \frac{6x{sin(x)}^{x}cos^{4}(x)}{sin^{4}(x)} + 3x^{2}{sin(x)}^{x} - 2x{sin(x)}^{x}\\ \end{split}\end{equation} \]

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