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。
\[ \begin{equation}\begin{split}【1/1】求函数{{(x + 5)}^{(2x + 3)}}^{(4x + 5)} 关于 x 的 2 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {(x + 5)^{(2x + 3)}}^{(4x + 5)}\right)}{dx}\\=&({(x + 5)^{(2x + 3)}}^{(4x + 5)}((4 + 0)ln((x + 5)^{(2x + 3)}) + \frac{(4x + 5)(((x + 5)^{(2x + 3)}((2 + 0)ln(x + 5) + \frac{(2x + 3)(1 + 0)}{(x + 5)})))}{((x + 5)^{(2x + 3)})}))\\=&4{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln((x + 5)^{(2x + 3)}) + 8x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5) + 10(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5) + \frac{8x^{2}(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)} + \frac{22x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)} + \frac{15(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( 4{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln((x + 5)^{(2x + 3)}) + 8x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5) + 10(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5) + \frac{8x^{2}(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)} + \frac{22x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)} + \frac{15(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)}\right)}{dx}\\=&4({(x + 5)^{(2x + 3)}}^{(4x + 5)}((4 + 0)ln((x + 5)^{(2x + 3)}) + \frac{(4x + 5)(((x + 5)^{(2x + 3)}((2 + 0)ln(x + 5) + \frac{(2x + 3)(1 + 0)}{(x + 5)})))}{((x + 5)^{(2x + 3)})}))ln((x + 5)^{(2x + 3)}) + \frac{4{(x + 5)^{(2x + 3)}}^{(4x + 5)}((x + 5)^{(2x + 3)}((2 + 0)ln(x + 5) + \frac{(2x + 3)(1 + 0)}{(x + 5)}))}{((x + 5)^{(2x + 3)})} + 8(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5) + 8x((x + 5)^{(4x + 6)}((4 + 0)ln(x + 5) + \frac{(4x + 6)(1 + 0)}{(x + 5)})){(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5) + 8x(x + 5)^{(4x + 6)}({(x + 5)^{(2x + 3)}}^{(4x + 5)}((4 + 0)ln((x + 5)^{(2x + 3)}) + \frac{(4x + 5)(((x + 5)^{(2x + 3)}((2 + 0)ln(x + 5) + \frac{(2x + 3)(1 + 0)}{(x + 5)})))}{((x + 5)^{(2x + 3)})}))ln(x + 5) + \frac{8x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}(1 + 0)}{(x + 5)} + 10((x + 5)^{(4x + 6)}((4 + 0)ln(x + 5) + \frac{(4x + 6)(1 + 0)}{(x + 5)})){(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5) + 10(x + 5)^{(4x + 6)}({(x + 5)^{(2x + 3)}}^{(4x + 5)}((4 + 0)ln((x + 5)^{(2x + 3)}) + \frac{(4x + 5)(((x + 5)^{(2x + 3)}((2 + 0)ln(x + 5) + \frac{(2x + 3)(1 + 0)}{(x + 5)})))}{((x + 5)^{(2x + 3)})}))ln(x + 5) + \frac{10(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}(1 + 0)}{(x + 5)} + 8(\frac{-(1 + 0)}{(x + 5)^{2}})x^{2}(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)} + \frac{8*2x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)} + \frac{8x^{2}((x + 5)^{(4x + 6)}((4 + 0)ln(x + 5) + \frac{(4x + 6)(1 + 0)}{(x + 5)})){(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)} + \frac{8x^{2}(x + 5)^{(4x + 6)}({(x + 5)^{(2x + 3)}}^{(4x + 5)}((4 + 0)ln((x + 5)^{(2x + 3)}) + \frac{(4x + 5)(((x + 5)^{(2x + 3)}((2 + 0)ln(x + 5) + \frac{(2x + 3)(1 + 0)}{(x + 5)})))}{((x + 5)^{(2x + 3)})}))}{(x + 5)} + 22(\frac{-(1 + 0)}{(x + 5)^{2}})x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)} + \frac{22(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)} + \frac{22x((x + 5)^{(4x + 6)}((4 + 0)ln(x + 5) + \frac{(4x + 6)(1 + 0)}{(x + 5)})){(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)} + \frac{22x(x + 5)^{(4x + 6)}({(x + 5)^{(2x + 3)}}^{(4x + 5)}((4 + 0)ln((x + 5)^{(2x + 3)}) + \frac{(4x + 5)(((x + 5)^{(2x + 3)}((2 + 0)ln(x + 5) + \frac{(2x + 3)(1 + 0)}{(x + 5)})))}{((x + 5)^{(2x + 3)})}))}{(x + 5)} + 15(\frac{-(1 + 0)}{(x + 5)^{2}})(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)} + \frac{15((x + 5)^{(4x + 6)}((4 + 0)ln(x + 5) + \frac{(4x + 6)(1 + 0)}{(x + 5)})){(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)} + \frac{15(x + 5)^{(4x + 6)}({(x + 5)^{(2x + 3)}}^{(4x + 5)}((4 + 0)ln((x + 5)^{(2x + 3)}) + \frac{(4x + 5)(((x + 5)^{(2x + 3)}((2 + 0)ln(x + 5) + \frac{(2x + 3)(1 + 0)}{(x + 5)})))}{((x + 5)^{(2x + 3)})}))}{(x + 5)}\\=&16{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln^{2}((x + 5)^{(2x + 3)}) + 32x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)ln((x + 5)^{(2x + 3)}) + 40(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)ln((x + 5)^{(2x + 3)}) + \frac{32x^{2}(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln((x + 5)^{(2x + 3)})}{(x + 5)} + \frac{88x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln((x + 5)^{(2x + 3)})}{(x + 5)} + \frac{60(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln((x + 5)^{(2x + 3)})}{(x + 5)} + 40{(x + 5)^{(2x + 3)}}^{(4x + 5)}(x + 5)^{(4x + 6)}ln((x + 5)^{(2x + 3)})ln(x + 5) + \frac{64x^{2}(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)}{(x + 5)} + \frac{120(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)}{(x + 5)} + 8(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5) + 32x{(x + 5)^{(2x + 3)}}^{(4x + 5)}(x + 5)^{(4x + 6)}ln((x + 5)^{(2x + 3)})ln(x + 5) + \frac{176x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)}{(x + 5)} + \frac{128x^{3}(x + 5)^{(-8x - 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)}{(x + 5)} + 32x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln^{2}(x + 5) + 64x^{2}(x + 5)^{(-8x - 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln^{2}(x + 5) + 80x(x + 5)^{(-8x - 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln^{2}(x + 5) + \frac{432x^{2}(x + 5)^{(-8x - 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)}{(x + 5)} + \frac{340x(x + 5)^{(-8x - 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)}{(x + 5)} + \frac{80x^{2}(x + 5)^{(8x + 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)}{(x + 5)} + \frac{340x(x + 5)^{(8x + 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)}{(x + 5)} + 40(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln^{2}(x + 5) + \frac{300(x + 5)^{(8x + 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln(x + 5)}{(x + 5)} + 8{(x + 5)^{(2x + 3)}}^{(4x + 5)}(x + 5)^{(4x + 6)}ln(x + 5) + 80x(x + 5)^{(8x + 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln^{2}(x + 5) + 100(x + 5)^{(8x + 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}ln^{2}(x + 5) + \frac{32x^{2}{(x + 5)^{(2x + 3)}}^{(4x + 5)}(x + 5)^{(4x + 6)}ln((x + 5)^{(2x + 3)})}{(x + 5)} + \frac{88x{(x + 5)^{(2x + 3)}}^{(4x + 5)}(x + 5)^{(4x + 6)}ln((x + 5)^{(2x + 3)})}{(x + 5)} + \frac{60{(x + 5)^{(2x + 3)}}^{(4x + 5)}(x + 5)^{(4x + 6)}ln((x + 5)^{(2x + 3)})}{(x + 5)} + \frac{225(x + 5)^{(8x + 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{352x^{3}(x + 5)^{(-8x - 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{604x^{2}(x + 5)^{(-8x - 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{64x^{4}(x + 5)^{(-8x - 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{330x(x + 5)^{(-8x - 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{120x^{2}(x + 5)^{(8x + 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{330x(x + 5)^{(8x + 12)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{170x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{8x{(x + 5)^{(2x + 3)}}^{(4x + 5)}(x + 5)^{(4x + 6)}}{(x + 5)} + \frac{24x(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)} + \frac{75(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{12{(x + 5)^{(2x + 3)}}^{(4x + 5)}(x + 5)^{(4x + 6)}}{(x + 5)} + \frac{128x^{2}(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{32x^{3}(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)^{2}} + \frac{32(x + 5)^{(4x + 6)}{(x + 5)^{(2x + 3)}}^{(4x + 5)}}{(x + 5)}\\ \end{split}\end{equation} \]

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