本次共计算 1 个题目：每一题对 x 求 3 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数xsin(x)cos(x)ln(x){e}^{x} 关于 x 的 3 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = x{e}^{x}ln(x)sin(x)cos(x)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( x{e}^{x}ln(x)sin(x)cos(x)\right)}{dx}\\=&{e}^{x}ln(x)sin(x)cos(x) + x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)sin(x)cos(x) + \frac{x{e}^{x}sin(x)cos(x)}{(x)} + x{e}^{x}ln(x)cos(x)cos(x) + x{e}^{x}ln(x)sin(x)*-sin(x)\\=&{e}^{x}ln(x)sin(x)cos(x) + x{e}^{x}ln(x)sin(x)cos(x) + {e}^{x}sin(x)cos(x) + x{e}^{x}ln(x)cos^{2}(x) - x{e}^{x}ln(x)sin^{2}(x)\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( {e}^{x}ln(x)sin(x)cos(x) + x{e}^{x}ln(x)sin(x)cos(x) + {e}^{x}sin(x)cos(x) + x{e}^{x}ln(x)cos^{2}(x) - x{e}^{x}ln(x)sin^{2}(x)\right)}{dx}\\=&({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)sin(x)cos(x) + \frac{{e}^{x}sin(x)cos(x)}{(x)} + {e}^{x}ln(x)cos(x)cos(x) + {e}^{x}ln(x)sin(x)*-sin(x) + {e}^{x}ln(x)sin(x)cos(x) + x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)sin(x)cos(x) + \frac{x{e}^{x}sin(x)cos(x)}{(x)} + x{e}^{x}ln(x)cos(x)cos(x) + x{e}^{x}ln(x)sin(x)*-sin(x) + ({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(x)cos(x) + {e}^{x}cos(x)cos(x) + {e}^{x}sin(x)*-sin(x) + {e}^{x}ln(x)cos^{2}(x) + x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)cos^{2}(x) + \frac{x{e}^{x}cos^{2}(x)}{(x)} + x{e}^{x}ln(x)*-2cos(x)sin(x) - {e}^{x}ln(x)sin^{2}(x) - x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)sin^{2}(x) - \frac{x{e}^{x}sin^{2}(x)}{(x)} - x{e}^{x}ln(x)*2sin(x)cos(x)\\=&2{e}^{x}ln(x)sin(x)cos(x) + \frac{{e}^{x}sin(x)cos(x)}{x} + 2{e}^{x}ln(x)cos^{2}(x) - 2{e}^{x}ln(x)sin^{2}(x) - 3x{e}^{x}ln(x)sin(x)cos(x) + 2{e}^{x}sin(x)cos(x) + 2x{e}^{x}ln(x)cos^{2}(x) + 2{e}^{x}cos^{2}(x) - 2{e}^{x}sin^{2}(x) - 2x{e}^{x}ln(x)sin^{2}(x)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2{e}^{x}ln(x)sin(x)cos(x) + \frac{{e}^{x}sin(x)cos(x)}{x} + 2{e}^{x}ln(x)cos^{2}(x) - 2{e}^{x}ln(x)sin^{2}(x) - 3x{e}^{x}ln(x)sin(x)cos(x) + 2{e}^{x}sin(x)cos(x) + 2x{e}^{x}ln(x)cos^{2}(x) + 2{e}^{x}cos^{2}(x) - 2{e}^{x}sin^{2}(x) - 2x{e}^{x}ln(x)sin^{2}(x)\right)}{dx}\\=&2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)sin(x)cos(x) + \frac{2{e}^{x}sin(x)cos(x)}{(x)} + 2{e}^{x}ln(x)cos(x)cos(x) + 2{e}^{x}ln(x)sin(x)*-sin(x) + \frac{-{e}^{x}sin(x)cos(x)}{x^{2}} + \frac{({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(x)cos(x)}{x} + \frac{{e}^{x}cos(x)cos(x)}{x} + \frac{{e}^{x}sin(x)*-sin(x)}{x} + 2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)cos^{2}(x) + \frac{2{e}^{x}cos^{2}(x)}{(x)} + 2{e}^{x}ln(x)*-2cos(x)sin(x) - 2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)sin^{2}(x) - \frac{2{e}^{x}sin^{2}(x)}{(x)} - 2{e}^{x}ln(x)*2sin(x)cos(x) - 3{e}^{x}ln(x)sin(x)cos(x) - 3x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)sin(x)cos(x) - \frac{3x{e}^{x}sin(x)cos(x)}{(x)} - 3x{e}^{x}ln(x)cos(x)cos(x) - 3x{e}^{x}ln(x)sin(x)*-sin(x) + 2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(x)cos(x) + 2{e}^{x}cos(x)cos(x) + 2{e}^{x}sin(x)*-sin(x) + 2{e}^{x}ln(x)cos^{2}(x) + 2x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)cos^{2}(x) + \frac{2x{e}^{x}cos^{2}(x)}{(x)} + 2x{e}^{x}ln(x)*-2cos(x)sin(x) + 2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))cos^{2}(x) + 2{e}^{x}*-2cos(x)sin(x) - 2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin^{2}(x) - 2{e}^{x}*2sin(x)cos(x) - 2{e}^{x}ln(x)sin^{2}(x) - 2x({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)sin^{2}(x) - \frac{2x{e}^{x}sin^{2}(x)}{(x)} - 2x{e}^{x}ln(x)*2sin(x)cos(x)\\=&-9{e}^{x}ln(x)sin(x)cos(x) + \frac{3{e}^{x}sin(x)cos(x)}{x} + 6{e}^{x}ln(x)cos^{2}(x) - \frac{{e}^{x}sin(x)cos(x)}{x^{2}} + \frac{3{e}^{x}cos^{2}(x)}{x} - \frac{3{e}^{x}sin^{2}(x)}{x} - 6{e}^{x}ln(x)sin^{2}(x) - 11x{e}^{x}ln(x)sin(x)cos(x) - 9{e}^{x}sin(x)cos(x) - x{e}^{x}ln(x)cos^{2}(x) + 6{e}^{x}cos^{2}(x) - 6{e}^{x}sin^{2}(x) + x{e}^{x}ln(x)sin^{2}(x)\\ \end{split}\end{equation} \]

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