本次共计算 1 个题目：每一题对 x 求 2 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数sin({3}^{x}) + {4}^{sin({4}^{x})} + {{x}^{5}}^{x} + {{6}^{x}}^{x} + {{x}^{7}}^{sin({8}^{x})} 关于 x 的 2 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = sin({3}^{x}) + {4}^{sin({4}^{x})} + {x^{5}}^{x} + {{6}^{x}}^{x} + {x^{7}}^{sin({8}^{x})}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin({3}^{x}) + {4}^{sin({4}^{x})} + {x^{5}}^{x} + {{6}^{x}}^{x} + {x^{7}}^{sin({8}^{x})}\right)}{dx}\\=&cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) + ({4}^{sin({4}^{x})}((cos({4}^{x})({4}^{x}((1)ln(4) + \frac{(x)(0)}{(4)})))ln(4) + \frac{(sin({4}^{x}))(0)}{(4)})) + ({x^{5}}^{x}((1)ln(x^{5}) + \frac{(x)(5x^{4})}{(x^{5})})) + ({{6}^{x}}^{x}((1)ln({6}^{x}) + \frac{(x)(({6}^{x}((1)ln(6) + \frac{(x)(0)}{(6)})))}{({6}^{x})})) + ({x^{7}}^{sin({8}^{x})}((cos({8}^{x})({8}^{x}((1)ln(8) + \frac{(x)(0)}{(8)})))ln(x^{7}) + \frac{(sin({8}^{x}))(7x^{6})}{(x^{7})}))\\=&{3}^{x}ln(3)cos({3}^{x}) + {4}^{x}{4}^{sin({4}^{x})}ln^{2}(4)cos({4}^{x}) + {x^{5}}^{x}ln(x^{5}) + {{6}^{x}}^{x}ln({6}^{x}) + {8}^{x}{x^{7}}^{sin({8}^{x})}ln(x^{7})ln(8)cos({8}^{x}) + x{{6}^{x}}^{x}ln(6) + 5{x^{5}}^{x} + \frac{7{x^{7}}^{sin({8}^{x})}sin({8}^{x})}{x}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( {3}^{x}ln(3)cos({3}^{x}) + {4}^{x}{4}^{sin({4}^{x})}ln^{2}(4)cos({4}^{x}) + {x^{5}}^{x}ln(x^{5}) + {{6}^{x}}^{x}ln({6}^{x}) + {8}^{x}{x^{7}}^{sin({8}^{x})}ln(x^{7})ln(8)cos({8}^{x}) + x{{6}^{x}}^{x}ln(6) + 5{x^{5}}^{x} + \frac{7{x^{7}}^{sin({8}^{x})}sin({8}^{x})}{x}\right)}{dx}\\=&({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)}))ln(3)cos({3}^{x}) + \frac{{3}^{x}*0cos({3}^{x})}{(3)} + {3}^{x}ln(3)*-sin({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) + ({4}^{x}((1)ln(4) + \frac{(x)(0)}{(4)})){4}^{sin({4}^{x})}ln^{2}(4)cos({4}^{x}) + {4}^{x}({4}^{sin({4}^{x})}((cos({4}^{x})({4}^{x}((1)ln(4) + \frac{(x)(0)}{(4)})))ln(4) + \frac{(sin({4}^{x}))(0)}{(4)}))ln^{2}(4)cos({4}^{x}) + \frac{{4}^{x}{4}^{sin({4}^{x})}*2ln(4)*0cos({4}^{x})}{(4)} + {4}^{x}{4}^{sin({4}^{x})}ln^{2}(4)*-sin({4}^{x})({4}^{x}((1)ln(4) + \frac{(x)(0)}{(4)})) + ({x^{5}}^{x}((1)ln(x^{5}) + \frac{(x)(5x^{4})}{(x^{5})}))ln(x^{5}) + \frac{{x^{5}}^{x}*5x^{4}}{(x^{5})} + ({{6}^{x}}^{x}((1)ln({6}^{x}) + \frac{(x)(({6}^{x}((1)ln(6) + \frac{(x)(0)}{(6)})))}{({6}^{x})}))ln({6}^{x}) + \frac{{{6}^{x}}^{x}({6}^{x}((1)ln(6) + \frac{(x)(0)}{(6)}))}{({6}^{x})} + ({8}^{x}((1)ln(8) + \frac{(x)(0)}{(8)})){x^{7}}^{sin({8}^{x})}ln(x^{7})ln(8)cos({8}^{x}) + {8}^{x}({x^{7}}^{sin({8}^{x})}((cos({8}^{x})({8}^{x}((1)ln(8) + \frac{(x)(0)}{(8)})))ln(x^{7}) + \frac{(sin({8}^{x}))(7x^{6})}{(x^{7})}))ln(x^{7})ln(8)cos({8}^{x}) + \frac{{8}^{x}{x^{7}}^{sin({8}^{x})}*7x^{6}ln(8)cos({8}^{x})}{(x^{7})} + \frac{{8}^{x}{x^{7}}^{sin({8}^{x})}ln(x^{7})*0cos({8}^{x})}{(8)} + {8}^{x}{x^{7}}^{sin({8}^{x})}ln(x^{7})ln(8)*-sin({8}^{x})({8}^{x}((1)ln(8) + \frac{(x)(0)}{(8)})) + {{6}^{x}}^{x}ln(6) + x({{6}^{x}}^{x}((1)ln({6}^{x}) + \frac{(x)(({6}^{x}((1)ln(6) + \frac{(x)(0)}{(6)})))}{({6}^{x})}))ln(6) + \frac{x{{6}^{x}}^{x}*0}{(6)} + 5({x^{5}}^{x}((1)ln(x^{5}) + \frac{(x)(5x^{4})}{(x^{5})})) + \frac{7*-{x^{7}}^{sin({8}^{x})}sin({8}^{x})}{x^{2}} + \frac{7({x^{7}}^{sin({8}^{x})}((cos({8}^{x})({8}^{x}((1)ln(8) + \frac{(x)(0)}{(8)})))ln(x^{7}) + \frac{(sin({8}^{x}))(7x^{6})}{(x^{7})}))sin({8}^{x})}{x} + \frac{7{x^{7}}^{sin({8}^{x})}cos({8}^{x})({8}^{x}((1)ln(8) + \frac{(x)(0)}{(8)}))}{x}\\=&{3}^{x}ln^{2}(3)cos({3}^{x}) - {3}^{(2x)}ln^{2}(3)sin({3}^{x}) + {4}^{x}{4}^{sin({4}^{x})}ln^{3}(4)cos({4}^{x}) + {4}^{(2x)}{4}^{sin({4}^{x})}ln^{4}(4)cos^{2}({4}^{x}) - {4}^{sin({4}^{x})}{4}^{(2x)}ln^{3}(4)sin({4}^{x}) + {x^{5}}^{x}ln^{2}(x^{5}) + {{6}^{x}}^{x}ln^{2}({6}^{x}) + \frac{7{x^{7}}^{sin({8}^{x})}{8}^{x}ln(8)cos({8}^{x})}{x} + 10{x^{5}}^{x}ln(x^{5}) + x{{6}^{x}}^{x}ln(6)ln({6}^{x}) - {x^{7}}^{sin({8}^{x})}{8}^{(2x)}ln^{2}(8)ln(x^{7})sin({8}^{x}) + {8}^{x}{x^{7}}^{sin({8}^{x})}ln^{2}(8)ln(x^{7})cos({8}^{x}) + {8}^{(2x)}{x^{7}}^{sin({8}^{x})}ln^{2}(8)ln^{2}(x^{7})cos^{2}({8}^{x}) + \frac{7{x^{7}}^{sin({8}^{x})}{8}^{x}ln(x^{7})ln(8)sin({8}^{x})cos({8}^{x})}{x} - \frac{7{x^{7}}^{sin({8}^{x})}sin({8}^{x})}{x^{2}} + 2{{6}^{x}}^{x}ln(6) + x{{6}^{x}}^{x}ln({6}^{x})ln(6) + x^{2}{{6}^{x}}^{x}ln^{2}(6) + 25{x^{5}}^{x} + \frac{7 * {8}^{x}{x^{7}}^{sin({8}^{x})}ln(8)ln(x^{7})sin({8}^{x})cos({8}^{x})}{x} + \frac{49{x^{7}}^{sin({8}^{x})}sin^{2}({8}^{x})}{x^{2}} + \frac{7 * {8}^{x}{x^{7}}^{sin({8}^{x})}ln(8)cos({8}^{x})}{x} + \frac{5{x^{5}}^{x}}{x}\\ \end{split}\end{equation} \]

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