本次共计算 2 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/2】求函数th(xx + 5x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = th(x^{2} + 5x)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( th(x^{2} + 5x)\right)}{dx}\\=&(1 - th^{2}(x^{2} + 5x))(2x + 5)\\=& - 2xth^{2}(x^{2} + 5x) + 2x - 5th^{2}(x^{2} + 5x) + 5\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( - 2xth^{2}(x^{2} + 5x) + 2x - 5th^{2}(x^{2} + 5x) + 5\right)}{dx}\\=& - 2th^{2}(x^{2} + 5x) - 2x*2th(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) + 2 - 5*2th(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) + 0\\=& - 2th^{2}(x^{2} + 5x) - 8x^{2}th(x^{2} + 5x) + 40xth^{3}(x^{2} + 5x) + 8x^{2}th^{3}(x^{2} + 5x) - 40xth(x^{2} + 5x) - 50th(x^{2} + 5x) + 50th^{3}(x^{2} + 5x) + 2\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - 2th^{2}(x^{2} + 5x) - 8x^{2}th(x^{2} + 5x) + 40xth^{3}(x^{2} + 5x) + 8x^{2}th^{3}(x^{2} + 5x) - 40xth(x^{2} + 5x) - 50th(x^{2} + 5x) + 50th^{3}(x^{2} + 5x) + 2\right)}{dx}\\=& - 2*2th(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) - 8*2xth(x^{2} + 5x) - 8x^{2}(1 - th^{2}(x^{2} + 5x))(2x + 5) + 40th^{3}(x^{2} + 5x) + 40x*3th^{2}(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) + 8*2xth^{3}(x^{2} + 5x) + 8x^{2}*3th^{2}(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) - 40th(x^{2} + 5x) - 40x(1 - th^{2}(x^{2} + 5x))(2x + 5) - 50(1 - th^{2}(x^{2} + 5x))(2x + 5) + 50*3th^{2}(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) + 0\\=& - 24xth(x^{2} + 5x) - 60th(x^{2} + 5x) + 24xth^{3}(x^{2} + 5x) + 60th^{3}(x^{2} + 5x) + 480x^{2}th^{2}(x^{2} + 5x) - 360x^{2}th^{4}(x^{2} + 5x) - 900xth^{4}(x^{2} + 5x) + 64x^{3}th^{2}(x^{2} + 5x) - 48x^{3}th^{4}(x^{2} + 5x) + 1200xth^{2}(x^{2} + 5x) - 120x^{2} - 16x^{3} - 300x - 750th^{4}(x^{2} + 5x) + 1000th^{2}(x^{2} + 5x) - 250\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( - 24xth(x^{2} + 5x) - 60th(x^{2} + 5x) + 24xth^{3}(x^{2} + 5x) + 60th^{3}(x^{2} + 5x) + 480x^{2}th^{2}(x^{2} + 5x) - 360x^{2}th^{4}(x^{2} + 5x) - 900xth^{4}(x^{2} + 5x) + 64x^{3}th^{2}(x^{2} + 5x) - 48x^{3}th^{4}(x^{2} + 5x) + 1200xth^{2}(x^{2} + 5x) - 120x^{2} - 16x^{3} - 300x - 750th^{4}(x^{2} + 5x) + 1000th^{2}(x^{2} + 5x) - 250\right)}{dx}\\=& - 24th(x^{2} + 5x) - 24x(1 - th^{2}(x^{2} + 5x))(2x + 5) - 60(1 - th^{2}(x^{2} + 5x))(2x + 5) + 24th^{3}(x^{2} + 5x) + 24x*3th^{2}(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) + 60*3th^{2}(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) + 480*2xth^{2}(x^{2} + 5x) + 480x^{2}*2th(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) - 360*2xth^{4}(x^{2} + 5x) - 360x^{2}*4th^{3}(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) - 900th^{4}(x^{2} + 5x) - 900x*4th^{3}(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) + 64*3x^{2}th^{2}(x^{2} + 5x) + 64x^{3}*2th(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) - 48*3x^{2}th^{4}(x^{2} + 5x) - 48x^{3}*4th^{3}(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) + 1200th^{2}(x^{2} + 5x) + 1200x*2th(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) - 120*2x - 16*3x^{2} - 300 - 750*4th^{3}(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) + 1000*2th(x^{2} + 5x)(1 - th^{2}(x^{2} + 5x))(2x + 5) + 0\\=&9976th(x^{2} + 5x) + 384x^{2}th^{2}(x^{2} + 5x) - 288x^{2}th^{4}(x^{2} + 5x) - 1440xth^{4}(x^{2} + 5x) + 2560x^{3}th(x^{2} + 5x) + 24000xth^{5}(x^{2} + 5x) + 1920xth^{2}(x^{2} + 5x) - 1800th^{4}(x^{2} + 5x) - 24976th^{3}(x^{2} + 5x) - 6400x^{3}th^{3}(x^{2} + 5x) - 40000xth^{3}(x^{2} + 5x) - 24000x^{2}th^{3}(x^{2} + 5x) + 2400th^{2}(x^{2} + 5x) + 9600x^{2}th(x^{2} + 5x) + 15000th^{5}(x^{2} + 5x) + 3840x^{3}th^{5}(x^{2} + 5x) + 256x^{4}th(x^{2} + 5x) + 14400x^{2}th^{5}(x^{2} + 5x) - 640x^{4}th^{3}(x^{2} + 5x) + 384x^{4}th^{5}(x^{2} + 5x) + 16000xth(x^{2} + 5x) - 96x^{2} - 480x - 600\\ \end{split}\end{equation} \]

\[ \begin{equation}\begin{split}【2/2】求函数tanh(xx + 5x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = tanh(x^{2} + 5x)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( tanh(x^{2} + 5x)\right)}{dx}\\=&sech^{2}(x^{2} + 5x)(2x + 5)\\=&2xsech^{2}(x^{2} + 5x) + 5sech^{2}(x^{2} + 5x)\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( 2xsech^{2}(x^{2} + 5x) + 5sech^{2}(x^{2} + 5x)\right)}{dx}\\=&2sech^{2}(x^{2} + 5x) + 2x*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) + 5*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5)\\=&2sech^{2}(x^{2} + 5x) - 8x^{2}tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 40xtanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 50tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2sech^{2}(x^{2} + 5x) - 8x^{2}tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 40xtanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 50tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x)\right)}{dx}\\=&2*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) - 8*2xtanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 8x^{2}sech^{2}(x^{2} + 5x)(2x + 5)sech^{2}(x^{2} + 5x) - 8x^{2}tanh(x^{2} + 5x)*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) - 40tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 40xsech^{2}(x^{2} + 5x)(2x + 5)sech^{2}(x^{2} + 5x) - 40xtanh(x^{2} + 5x)*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) - 50sech^{2}(x^{2} + 5x)(2x + 5)sech^{2}(x^{2} + 5x) - 50tanh(x^{2} + 5x)*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5)\\=&-24xtanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 60tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 16x^{3}sech^{4}(x^{2} + 5x) - 120x^{2}sech^{4}(x^{2} + 5x) + 32x^{3}tanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) + 240x^{2}tanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 300xsech^{4}(x^{2} + 5x) + 600xtanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 250sech^{4}(x^{2} + 5x) + 500tanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( -24xtanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 60tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 16x^{3}sech^{4}(x^{2} + 5x) - 120x^{2}sech^{4}(x^{2} + 5x) + 32x^{3}tanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) + 240x^{2}tanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 300xsech^{4}(x^{2} + 5x) + 600xtanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 250sech^{4}(x^{2} + 5x) + 500tanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x)\right)}{dx}\\=&-24tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 24xsech^{2}(x^{2} + 5x)(2x + 5)sech^{2}(x^{2} + 5x) - 24xtanh(x^{2} + 5x)*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) - 60sech^{2}(x^{2} + 5x)(2x + 5)sech^{2}(x^{2} + 5x) - 60tanh(x^{2} + 5x)*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) - 16*3x^{2}sech^{4}(x^{2} + 5x) - 16x^{3}*-4sech^{3}(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) - 120*2xsech^{4}(x^{2} + 5x) - 120x^{2}*-4sech^{3}(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) + 32*3x^{2}tanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) + 32x^{3}*2tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x)(2x + 5)sech^{2}(x^{2} + 5x) + 32x^{3}tanh^{2}(x^{2} + 5x)*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) + 240*2xtanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) + 240x^{2}*2tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x)(2x + 5)sech^{2}(x^{2} + 5x) + 240x^{2}tanh^{2}(x^{2} + 5x)*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) - 300sech^{4}(x^{2} + 5x) - 300x*-4sech^{3}(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) + 600tanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) + 600x*2tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x)(2x + 5)sech^{2}(x^{2} + 5x) + 600xtanh^{2}(x^{2} + 5x)*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) - 250*-4sech^{3}(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5) + 500*2tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x)(2x + 5)sech^{2}(x^{2} + 5x) + 500tanh^{2}(x^{2} + 5x)*-2sech(x^{2} + 5x)sech(x^{2} + 5x)tanh(x^{2} + 5x)(2x + 5)\\=&-24tanh(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 96x^{2}sech^{4}(x^{2} + 5x) - 480xsech^{4}(x^{2} + 5x) + 192x^{2}tanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) + 960xtanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 600sech^{4}(x^{2} + 5x) + 1200tanh^{2}(x^{2} + 5x)sech^{2}(x^{2} + 5x) + 256x^{4}tanh(x^{2} + 5x)sech^{4}(x^{2} + 5x) + 2560x^{3}tanh(x^{2} + 5x)sech^{4}(x^{2} + 5x) + 9600x^{2}tanh(x^{2} + 5x)sech^{4}(x^{2} + 5x) - 128x^{4}tanh^{3}(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 1280x^{3}tanh^{3}(x^{2} + 5x)sech^{2}(x^{2} + 5x) - 4800x^{2}tanh^{3}(x^{2} + 5x)sech^{2}(x^{2} + 5x) + 16000xtanh(x^{2} + 5x)sech^{4}(x^{2} + 5x) - 8000xtanh^{3}(x^{2} + 5x)sech^{2}(x^{2} + 5x) + 10000tanh(x^{2} + 5x)sech^{4}(x^{2} + 5x) - 5000tanh^{3}(x^{2} + 5x)sech^{2}(x^{2} + 5x)\\ \end{split}\end{equation} \]

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