本次共计算 1 个题目：每一题对 x 求 15 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{{2}^{x}}{({4}^{x} + 1)} 关于 x 的 15 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{{2}^{x}}{({4}^{x} + 1)}\\\\ &\color{blue}{函数的 15 阶导数：} \\=&\frac{87178291200 * {4}^{(15x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{15}} - \frac{647610163200 * {4}^{(14x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{14}} + \frac{2135868134400 * {4}^{(13x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{13}} - \frac{4115541830400 * {4}^{(12x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{12}} + \frac{5128989465600 * {4}^{(11x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{11}} - \frac{4321863786240 * {4}^{(10x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{10}} + \frac{2499572275200 * {4}^{(9x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{9}} - \frac{986406220800 * {4}^{(8x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{8}} + \frac{258776678160 * {4}^{(7x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{7}} - \frac{42870828000 * {4}^{(6x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{6}} + \frac{4096605240 * {4}^{(5x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{5}} - \frac{191785230 * {4}^{(4x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{4}} + \frac{3172262 * {4}^{(3x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{3}} - \frac{8192 * {4}^{(2x)}ln^{14}(4)ln(2)}{({4}^{x} + 1)^{2}} - \frac{6227020800 * {4}^{(14x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{14}} + \frac{43110144000 * {4}^{(13x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{13}} - \frac{131286355200 * {4}^{(12x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{12}} + \frac{230918688000 * {4}^{(11x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{11}} - \frac{258888389760 * {4}^{(10x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{10}} + \frac{192553320960 * {4}^{(9x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{9}} - \frac{95824048320 * {4}^{(8x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{8}} + \frac{31401976320 * {4}^{(7x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{7}} - \frac{6493807320 * {4}^{(6x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{6}} + \frac{781596816 * {4}^{(5x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{5}} - \frac{47155290 * {4}^{(4x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{4}} + \frac{1054690 * {4}^{(3x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{3}} - \frac{4096 * {4}^{(2x)}ln^{13}(4)ln^{2}(2)}{({4}^{x} + 1)^{2}} + \frac{479001600 * {4}^{(13x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{13}} - \frac{3073593600 * {4}^{(12x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{12}} + \frac{8582112000 * {4}^{(11x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{11}} - \frac{13651545600 * {4}^{(10x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{10}} + \frac{13596992640 * {4}^{(9x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{9}} - \frac{8772548400 * {4}^{(8x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{8}} + \frac{3663380160 * {4}^{(7x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{7}} - \frac{959719200 * {4}^{(6x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{6}} + \frac{147098424 * {4}^{(5x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{5}} - \frac{11526174 * {4}^{(4x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{4}} + \frac{350198 * {4}^{(3x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{3}} - \frac{2048 * {4}^{(2x)}ln^{12}(4)ln^{3}(2)}{({4}^{x} + 1)^{2}} - \frac{39916800 * {4}^{(12x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{12}} + \frac{235872000 * {4}^{(11x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{11}} - \frac{598752000 * {4}^{(10x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{10}} + \frac{851558400 * {4}^{(9x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{9}} - \frac{741620880 * {4}^{(8x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{8}} + \frac{405654480 * {4}^{(7x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{7}} - \frac{137299800 * {4}^{(6x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{6}} + \frac{27184080 * {4}^{(5x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{5}} - \frac{2794506 * {4}^{(4x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{4}} + \frac{116050 * {4}^{(3x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{3}} - \frac{1024 * {4}^{(2x)}ln^{11}(4)ln^{4}(2)}{({4}^{x} + 1)^{2}} + \frac{3628800 * {4}^{(11x)}ln^{10}(4)ln^{5}(2)}{({4}^{x} + 1)^{11}} - \frac{19595520 * {4}^{(10x)}ln^{10}(4)ln^{5}(2)}{({4}^{x} + 1)^{10}} + \frac{44755200 * {4}^{(9x)}ln^{10}(4)ln^{5}(2)}{({4}^{x} + 1)^{9}} - \frac{56095200 * {4}^{(8x)}ln^{10}(4)ln^{5}(2)}{({4}^{x} + 1)^{8}} + \frac{41837040 * {4}^{(7x)}ln^{10}(4)ln^{5}(2)}{({4}^{x} + 1)^{7}} - \frac{18799200 * {4}^{(6x)}ln^{10}(4)ln^{5}(2)}{({4}^{x} + 1)^{6}} + \frac{4900920 * {4}^{(5x)}ln^{10}(4)ln^{5}(2)}{({4}^{x} + 1)^{5}} - \frac{669870 * {4}^{(4x)}ln^{10}(4)ln^{5}(2)}{({4}^{x} + 1)^{4}} + \frac{38342 * {4}^{(3x)}ln^{10}(4)ln^{5}(2)}{({4}^{x} + 1)^{3}} - \frac{512 * {4}^{(2x)}ln^{10}(4)ln^{5}(2)}{({4}^{x} + 1)^{2}} - \frac{362880 * {4}^{(10x)}ln^{9}(4)ln^{6}(2)}{({4}^{x} + 1)^{10}} + \frac{1774080 * {4}^{(9x)}ln^{9}(4)ln^{6}(2)}{({4}^{x} + 1)^{9}} - \frac{3598560 * {4}^{(8x)}ln^{9}(4)ln^{6}(2)}{({4}^{x} + 1)^{8}} + \frac{3900960 * {4}^{(7x)}ln^{9}(4)ln^{6}(2)}{({4}^{x} + 1)^{7}} - \frac{2421720 * {4}^{(6x)}ln^{9}(4)ln^{6}(2)}{({4}^{x} + 1)^{6}} + \frac{853776 * {4}^{(5x)}ln^{9}(4)ln^{6}(2)}{({4}^{x} + 1)^{5}} - \frac{158010 * {4}^{(4x)}ln^{9}(4)ln^{6}(2)}{({4}^{x} + 1)^{4}} + \frac{12610 * {4}^{(3x)}ln^{9}(4)ln^{6}(2)}{({4}^{x} + 1)^{3}} - \frac{256 * {4}^{(2x)}ln^{9}(4)ln^{6}(2)}{({4}^{x} + 1)^{2}} + \frac{40320 * {4}^{(9x)}ln^{8}(4)ln^{7}(2)}{({4}^{x} + 1)^{9}} - \frac{176400 * {4}^{(8x)}ln^{8}(4)ln^{7}(2)}{({4}^{x} + 1)^{8}} + \frac{312480 * {4}^{(7x)}ln^{8}(4)ln^{7}(2)}{({4}^{x} + 1)^{7}} - \frac{285600 * {4}^{(6x)}ln^{8}(4)ln^{7}(2)}{({4}^{x} + 1)^{6}} + \frac{141624 * {4}^{(5x)}ln^{8}(4)ln^{7}(2)}{({4}^{x} + 1)^{5}} - \frac{36414 * {4}^{(4x)}ln^{8}(4)ln^{7}(2)}{({4}^{x} + 1)^{4}} + \frac{4118 * {4}^{(3x)}ln^{8}(4)ln^{7}(2)}{({4}^{x} + 1)^{3}} - \frac{128 * {4}^{(2x)}ln^{8}(4)ln^{7}(2)}{({4}^{x} + 1)^{2}} - \frac{5040 * {4}^{(8x)}ln^{7}(4)ln^{8}(2)}{({4}^{x} + 1)^{8}} + \frac{19440 * {4}^{(7x)}ln^{7}(4)ln^{8}(2)}{({4}^{x} + 1)^{7}} - \frac{29400 * {4}^{(6x)}ln^{7}(4)ln^{8}(2)}{({4}^{x} + 1)^{6}} + \frac{21840 * {4}^{(5x)}ln^{7}(4)ln^{8}(2)}{({4}^{x} + 1)^{5}} - \frac{8106 * {4}^{(4x)}ln^{7}(4)ln^{8}(2)}{({4}^{x} + 1)^{4}} + \frac{1330 * {4}^{(3x)}ln^{7}(4)ln^{8}(2)}{({4}^{x} + 1)^{3}} - \frac{64 * {4}^{(2x)}ln^{7}(4)ln^{8}(2)}{({4}^{x} + 1)^{2}} + \frac{720 * {4}^{(7x)}ln^{6}(4)ln^{9}(2)}{({4}^{x} + 1)^{7}} - \frac{2400 * {4}^{(6x)}ln^{6}(4)ln^{9}(2)}{({4}^{x} + 1)^{6}} + \frac{3000 * {4}^{(5x)}ln^{6}(4)ln^{9}(2)}{({4}^{x} + 1)^{5}} - \frac{1710 * {4}^{(4x)}ln^{6}(4)ln^{9}(2)}{({4}^{x} + 1)^{4}} + \frac{422 * {4}^{(3x)}ln^{6}(4)ln^{9}(2)}{({4}^{x} + 1)^{3}} - \frac{32 * {4}^{(2x)}ln^{6}(4)ln^{9}(2)}{({4}^{x} + 1)^{2}} - \frac{120 * {4}^{(6x)}ln^{5}(4)ln^{10}(2)}{({4}^{x} + 1)^{6}} + \frac{336 * {4}^{(5x)}ln^{5}(4)ln^{10}(2)}{({4}^{x} + 1)^{5}} - \frac{330 * {4}^{(4x)}ln^{5}(4)ln^{10}(2)}{({4}^{x} + 1)^{4}} + \frac{130 * {4}^{(3x)}ln^{5}(4)ln^{10}(2)}{({4}^{x} + 1)^{3}} - \frac{16 * {4}^{(2x)}ln^{5}(4)ln^{10}(2)}{({4}^{x} + 1)^{2}} + \frac{24 * {4}^{(5x)}ln^{4}(4)ln^{11}(2)}{({4}^{x} + 1)^{5}} - \frac{54 * {4}^{(4x)}ln^{4}(4)ln^{11}(2)}{({4}^{x} + 1)^{4}} + \frac{38 * {4}^{(3x)}ln^{4}(4)ln^{11}(2)}{({4}^{x} + 1)^{3}} - \frac{8 * {4}^{(2x)}ln^{4}(4)ln^{11}(2)}{({4}^{x} + 1)^{2}} - \frac{6 * {4}^{(4x)}ln^{3}(4)ln^{12}(2)}{({4}^{x} + 1)^{4}} + \frac{10 * {4}^{(3x)}ln^{3}(4)ln^{12}(2)}{({4}^{x} + 1)^{3}} - \frac{4 * {4}^{(2x)}ln^{3}(4)ln^{12}(2)}{({4}^{x} + 1)^{2}} + \frac{2 * {4}^{(3x)}ln^{2}(4)ln^{13}(2)}{({4}^{x} + 1)^{3}} - \frac{2 * {4}^{(2x)}ln^{2}(4)ln^{13}(2)}{({4}^{x} + 1)^{2}} - \frac{{4}^{(2x)}ln(4)ln^{14}(2)}{({4}^{x} + 1)^{2}} - \frac{16384 * {4}^{(2x)}ln^{15}(4)}{({4}^{x} + 1)^{2}} - \frac{1307674368000 * {4}^{(16x)}ln^{15}(4)}{({4}^{x} + 1)^{16}} + \frac{10374216652800 * {4}^{(15x)}ln^{15}(4)}{({4}^{x} + 1)^{15}} - \frac{36832828032000 * {4}^{(14x)}ln^{15}(4)}{({4}^{x} + 1)^{14}} + \frac{77152787712000 * {4}^{(13x)}ln^{15}(4)}{({4}^{x} + 1)^{13}} - \frac{105805386086400 * {4}^{(12x)}ln^{15}(4)}{({4}^{x} + 1)^{12}} + \frac{99637521984000 * {4}^{(11x)}ln^{15}(4)}{({4}^{x} + 1)^{11}} - \frac{65714788339200 * {4}^{(10x)}ln^{15}(4)}{({4}^{x} + 1)^{10}} + \frac{30387400243200 * {4}^{(9x)}ln^{15}(4)}{({4}^{x} + 1)^{9}} - \frac{9702686513520 * {4}^{(8x)}ln^{15}(4)}{({4}^{x} + 1)^{8}} + \frac{2068661715120 * {4}^{(7x)}ln^{15}(4)}{({4}^{x} + 1)^{7}} - \frac{277707994200 * {4}^{(6x)}ln^{15}(4)}{({4}^{x} + 1)^{6}} + \frac{21250167120 * {4}^{(5x)}ln^{15}(4)}{({4}^{x} + 1)^{5}} - \frac{776657706 * {4}^{(4x)}ln^{15}(4)}{({4}^{x} + 1)^{4}} + \frac{9533170 * {4}^{(3x)}ln^{15}(4)}{({4}^{x} + 1)^{3}} + \frac{{2}^{x}ln^{15}(2)}{({4}^{x} + 1)}\\ \end{split}\end{equation} \]

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