求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
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。
\[ \begin{equation}\begin{split}【1/2】求函数lg(th(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( lg(th(x))\right)}{dx}\\=&\frac{(1 - th^{2}(x))}{ln{10}(th(x))}\\=&\frac{1}{ln{10}th(x)} - \frac{th(x)}{ln{10}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{ln{10}th(x)} - \frac{th(x)}{ln{10}}\right)}{dx}\\=&\frac{-0}{ln^{2}{10}th(x)} + \frac{-(1 - th^{2}(x))}{ln{10}th^{2}(x)} - \frac{-0th(x)}{ln^{2}{10}} - \frac{(1 - th^{2}(x))}{ln{10}}\\=&\frac{-1}{ln{10}th^{2}(x)} + \frac{th^{2}(x)}{ln{10}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-1}{ln{10}th^{2}(x)} + \frac{th^{2}(x)}{ln{10}}\right)}{dx}\\=&\frac{--0}{ln^{2}{10}th^{2}(x)} - \frac{-2(1 - th^{2}(x))}{ln{10}th^{3}(x)} + \frac{-0th^{2}(x)}{ln^{2}{10}} + \frac{2th(x)(1 - th^{2}(x))}{ln{10}}\\=&\frac{2}{ln{10}th^{3}(x)} - \frac{2}{ln{10}th(x)} - \frac{2th^{3}(x)}{ln{10}} + \frac{2th(x)}{ln{10}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2}{ln{10}th^{3}(x)} - \frac{2}{ln{10}th(x)} - \frac{2th^{3}(x)}{ln{10}} + \frac{2th(x)}{ln{10}}\right)}{dx}\\=&\frac{2*-0}{ln^{2}{10}th^{3}(x)} + \frac{2*-3(1 - th^{2}(x))}{ln{10}th^{4}(x)} - \frac{2*-0}{ln^{2}{10}th(x)} - \frac{2*-(1 - th^{2}(x))}{ln{10}th^{2}(x)} - \frac{2*-0th^{3}(x)}{ln^{2}{10}} - \frac{2*3th^{2}(x)(1 - th^{2}(x))}{ln{10}} + \frac{2*-0th(x)}{ln^{2}{10}} + \frac{2(1 - th^{2}(x))}{ln{10}}\\=&\frac{-6}{ln{10}th^{4}(x)} + \frac{8}{ln{10}th^{2}(x)} + \frac{6th^{4}(x)}{ln{10}} - \frac{8th^{2}(x)}{ln{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/2】求函数lg(tanh(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( lg(tanh(x))\right)}{dx}\\=&\frac{sech^{2}(x)}{ln{10}(tanh(x))}\\=&\frac{sech^{2}(x)}{ln{10}tanh(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{sech^{2}(x)}{ln{10}tanh(x)}\right)}{dx}\\=&\frac{-0sech^{2}(x)}{ln^{2}{10}tanh(x)} + \frac{-sech^{2}(x)sech^{2}(x)}{ln{10}tanh^{2}(x)} + \frac{-2sech(x)sech(x)tanh(x)}{ln{10}tanh(x)}\\=&\frac{-sech^{4}(x)}{ln{10}tanh^{2}(x)} - \frac{2sech^{2}(x)}{ln{10}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-sech^{4}(x)}{ln{10}tanh^{2}(x)} - \frac{2sech^{2}(x)}{ln{10}}\right)}{dx}\\=&\frac{--0sech^{4}(x)}{ln^{2}{10}tanh^{2}(x)} - \frac{-2sech^{2}(x)sech^{4}(x)}{ln{10}tanh^{3}(x)} - \frac{-4sech^{3}(x)sech(x)tanh(x)}{ln{10}tanh^{2}(x)} - \frac{2*-0sech^{2}(x)}{ln^{2}{10}} - \frac{2*-2sech(x)sech(x)tanh(x)}{ln{10}}\\=&\frac{2sech^{6}(x)}{ln{10}tanh^{3}(x)} + \frac{4sech^{4}(x)}{ln{10}tanh(x)} + \frac{4tanh(x)sech^{2}(x)}{ln{10}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2sech^{6}(x)}{ln{10}tanh^{3}(x)} + \frac{4sech^{4}(x)}{ln{10}tanh(x)} + \frac{4tanh(x)sech^{2}(x)}{ln{10}}\right)}{dx}\\=&\frac{2*-0sech^{6}(x)}{ln^{2}{10}tanh^{3}(x)} + \frac{2*-3sech^{2}(x)sech^{6}(x)}{ln{10}tanh^{4}(x)} + \frac{2*-6sech^{5}(x)sech(x)tanh(x)}{ln{10}tanh^{3}(x)} + \frac{4*-0sech^{4}(x)}{ln^{2}{10}tanh(x)} + \frac{4*-sech^{2}(x)sech^{4}(x)}{ln{10}tanh^{2}(x)} + \frac{4*-4sech^{3}(x)sech(x)tanh(x)}{ln{10}tanh(x)} + \frac{4*-0tanh(x)sech^{2}(x)}{ln^{2}{10}} + \frac{4sech^{2}(x)sech^{2}(x)}{ln{10}} + \frac{4tanh(x)*-2sech(x)sech(x)tanh(x)}{ln{10}}\\=& - \frac{6sech^{8}(x)}{ln{10}tanh^{4}(x)} - \frac{16sech^{6}(x)}{ln{10}tanh^{2}(x)} - \frac{12sech^{4}(x)}{ln{10}} - \frac{8tanh^{2}(x)sech^{2}(x)}{ln{10}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/2】求函数lg(th(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( lg(th(x))\right)}{dx}\\=&\frac{(1 - th^{2}(x))}{ln{10}(th(x))}\\=&\frac{1}{ln{10}th(x)} - \frac{th(x)}{ln{10}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{ln{10}th(x)} - \frac{th(x)}{ln{10}}\right)}{dx}\\=&\frac{-0}{ln^{2}{10}th(x)} + \frac{-(1 - th^{2}(x))}{ln{10}th^{2}(x)} - \frac{-0th(x)}{ln^{2}{10}} - \frac{(1 - th^{2}(x))}{ln{10}}\\=&\frac{-1}{ln{10}th^{2}(x)} + \frac{th^{2}(x)}{ln{10}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-1}{ln{10}th^{2}(x)} + \frac{th^{2}(x)}{ln{10}}\right)}{dx}\\=&\frac{--0}{ln^{2}{10}th^{2}(x)} - \frac{-2(1 - th^{2}(x))}{ln{10}th^{3}(x)} + \frac{-0th^{2}(x)}{ln^{2}{10}} + \frac{2th(x)(1 - th^{2}(x))}{ln{10}}\\=&\frac{2}{ln{10}th^{3}(x)} - \frac{2}{ln{10}th(x)} - \frac{2th^{3}(x)}{ln{10}} + \frac{2th(x)}{ln{10}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2}{ln{10}th^{3}(x)} - \frac{2}{ln{10}th(x)} - \frac{2th^{3}(x)}{ln{10}} + \frac{2th(x)}{ln{10}}\right)}{dx}\\=&\frac{2*-0}{ln^{2}{10}th^{3}(x)} + \frac{2*-3(1 - th^{2}(x))}{ln{10}th^{4}(x)} - \frac{2*-0}{ln^{2}{10}th(x)} - \frac{2*-(1 - th^{2}(x))}{ln{10}th^{2}(x)} - \frac{2*-0th^{3}(x)}{ln^{2}{10}} - \frac{2*3th^{2}(x)(1 - th^{2}(x))}{ln{10}} + \frac{2*-0th(x)}{ln^{2}{10}} + \frac{2(1 - th^{2}(x))}{ln{10}}\\=&\frac{-6}{ln{10}th^{4}(x)} + \frac{8}{ln{10}th^{2}(x)} + \frac{6th^{4}(x)}{ln{10}} - \frac{8th^{2}(x)}{ln{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/2】求函数lg(tanh(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( lg(tanh(x))\right)}{dx}\\=&\frac{sech^{2}(x)}{ln{10}(tanh(x))}\\=&\frac{sech^{2}(x)}{ln{10}tanh(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{sech^{2}(x)}{ln{10}tanh(x)}\right)}{dx}\\=&\frac{-0sech^{2}(x)}{ln^{2}{10}tanh(x)} + \frac{-sech^{2}(x)sech^{2}(x)}{ln{10}tanh^{2}(x)} + \frac{-2sech(x)sech(x)tanh(x)}{ln{10}tanh(x)}\\=&\frac{-sech^{4}(x)}{ln{10}tanh^{2}(x)} - \frac{2sech^{2}(x)}{ln{10}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-sech^{4}(x)}{ln{10}tanh^{2}(x)} - \frac{2sech^{2}(x)}{ln{10}}\right)}{dx}\\=&\frac{--0sech^{4}(x)}{ln^{2}{10}tanh^{2}(x)} - \frac{-2sech^{2}(x)sech^{4}(x)}{ln{10}tanh^{3}(x)} - \frac{-4sech^{3}(x)sech(x)tanh(x)}{ln{10}tanh^{2}(x)} - \frac{2*-0sech^{2}(x)}{ln^{2}{10}} - \frac{2*-2sech(x)sech(x)tanh(x)}{ln{10}}\\=&\frac{2sech^{6}(x)}{ln{10}tanh^{3}(x)} + \frac{4sech^{4}(x)}{ln{10}tanh(x)} + \frac{4tanh(x)sech^{2}(x)}{ln{10}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2sech^{6}(x)}{ln{10}tanh^{3}(x)} + \frac{4sech^{4}(x)}{ln{10}tanh(x)} + \frac{4tanh(x)sech^{2}(x)}{ln{10}}\right)}{dx}\\=&\frac{2*-0sech^{6}(x)}{ln^{2}{10}tanh^{3}(x)} + \frac{2*-3sech^{2}(x)sech^{6}(x)}{ln{10}tanh^{4}(x)} + \frac{2*-6sech^{5}(x)sech(x)tanh(x)}{ln{10}tanh^{3}(x)} + \frac{4*-0sech^{4}(x)}{ln^{2}{10}tanh(x)} + \frac{4*-sech^{2}(x)sech^{4}(x)}{ln{10}tanh^{2}(x)} + \frac{4*-4sech^{3}(x)sech(x)tanh(x)}{ln{10}tanh(x)} + \frac{4*-0tanh(x)sech^{2}(x)}{ln^{2}{10}} + \frac{4sech^{2}(x)sech^{2}(x)}{ln{10}} + \frac{4tanh(x)*-2sech(x)sech(x)tanh(x)}{ln{10}}\\=& - \frac{6sech^{8}(x)}{ln{10}tanh^{4}(x)} - \frac{16sech^{6}(x)}{ln{10}tanh^{2}(x)} - \frac{12sech^{4}(x)}{ln{10}} - \frac{8tanh^{2}(x)sech^{2}(x)}{ln{10}}\\ \end{split}\end{equation} \]
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