本次共计算 1 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数(10ln(x + 1) + 30)sqrt(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( 10ln(x + 1)sqrt(x) + 30sqrt(x)\right)}{dx}\\=&\frac{10(1 + 0)sqrt(x)}{(x + 1)} + \frac{10ln(x + 1)*\frac{1}{2}}{(x)^{\frac{1}{2}}} + \frac{30*\frac{1}{2}}{(x)^{\frac{1}{2}}}\\=&\frac{10sqrt(x)}{(x + 1)} + \frac{5ln(x + 1)}{x^{\frac{1}{2}}} + \frac{15}{x^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{10sqrt(x)}{(x + 1)} + \frac{5ln(x + 1)}{x^{\frac{1}{2}}} + \frac{15}{x^{\frac{1}{2}}}\right)}{dx}\\=&10(\frac{-(1 + 0)}{(x + 1)^{2}})sqrt(x) + \frac{10*\frac{1}{2}}{(x + 1)(x)^{\frac{1}{2}}} + \frac{5*\frac{-1}{2}ln(x + 1)}{x^{\frac{3}{2}}} + \frac{5(1 + 0)}{x^{\frac{1}{2}}(x + 1)} + \frac{15*\frac{-1}{2}}{x^{\frac{3}{2}}}\\=&\frac{-10sqrt(x)}{(x + 1)^{2}} + \frac{10}{(x + 1)x^{\frac{1}{2}}} - \frac{5ln(x + 1)}{2x^{\frac{3}{2}}} - \frac{15}{2x^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-10sqrt(x)}{(x + 1)^{2}} + \frac{10}{(x + 1)x^{\frac{1}{2}}} - \frac{5ln(x + 1)}{2x^{\frac{3}{2}}} - \frac{15}{2x^{\frac{3}{2}}}\right)}{dx}\\=&-10(\frac{-2(1 + 0)}{(x + 1)^{3}})sqrt(x) - \frac{10*\frac{1}{2}}{(x + 1)^{2}(x)^{\frac{1}{2}}} + \frac{10(\frac{-(1 + 0)}{(x + 1)^{2}})}{x^{\frac{1}{2}}} + \frac{10*\frac{-1}{2}}{(x + 1)x^{\frac{3}{2}}} - \frac{5*\frac{-3}{2}ln(x + 1)}{2x^{\frac{5}{2}}} - \frac{5(1 + 0)}{2x^{\frac{3}{2}}(x + 1)} - \frac{15*\frac{-3}{2}}{2x^{\frac{5}{2}}}\\=&\frac{20sqrt(x)}{(x + 1)^{3}} - \frac{15}{(x + 1)^{2}x^{\frac{1}{2}}} - \frac{15}{2(x + 1)x^{\frac{3}{2}}} + \frac{15ln(x + 1)}{4x^{\frac{5}{2}}} + \frac{45}{4x^{\frac{5}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{20sqrt(x)}{(x + 1)^{3}} - \frac{15}{(x + 1)^{2}x^{\frac{1}{2}}} - \frac{15}{2(x + 1)x^{\frac{3}{2}}} + \frac{15ln(x + 1)}{4x^{\frac{5}{2}}} + \frac{45}{4x^{\frac{5}{2}}}\right)}{dx}\\=&20(\frac{-3(1 + 0)}{(x + 1)^{4}})sqrt(x) + \frac{20*\frac{1}{2}}{(x + 1)^{3}(x)^{\frac{1}{2}}} - \frac{15(\frac{-2(1 + 0)}{(x + 1)^{3}})}{x^{\frac{1}{2}}} - \frac{15*\frac{-1}{2}}{(x + 1)^{2}x^{\frac{3}{2}}} - \frac{15(\frac{-(1 + 0)}{(x + 1)^{2}})}{2x^{\frac{3}{2}}} - \frac{15*\frac{-3}{2}}{2(x + 1)x^{\frac{5}{2}}} + \frac{15*\frac{-5}{2}ln(x + 1)}{4x^{\frac{7}{2}}} + \frac{15(1 + 0)}{4x^{\frac{5}{2}}(x + 1)} + \frac{45*\frac{-5}{2}}{4x^{\frac{7}{2}}}\\=&\frac{-60sqrt(x)}{(x + 1)^{4}} + \frac{40}{(x + 1)^{3}x^{\frac{1}{2}}} + \frac{15}{(x + 1)^{2}x^{\frac{3}{2}}} + \frac{15}{(x + 1)x^{\frac{5}{2}}} - \frac{75ln(x + 1)}{8x^{\frac{7}{2}}} - \frac{225}{8x^{\frac{7}{2}}}\\ \end{split}\end{equation} \]

你的问题在这里没有得到解决？请到 热门难题 里面看看吧！