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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{(2{x}^{2} + 3{x}^{3})}{({x}^{2} + 3)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{2x^{2}}{(x^{2} + 3)} + \frac{3x^{3}}{(x^{2} + 3)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{2x^{2}}{(x^{2} + 3)} + \frac{3x^{3}}{(x^{2} + 3)}\right)}{dx}\\=&2(\frac{-(2x + 0)}{(x^{2} + 3)^{2}})x^{2} + \frac{2*2x}{(x^{2} + 3)} + 3(\frac{-(2x + 0)}{(x^{2} + 3)^{2}})x^{3} + \frac{3*3x^{2}}{(x^{2} + 3)}\\=&\frac{-4x^{3}}{(x^{2} + 3)^{2}} + \frac{4x}{(x^{2} + 3)} - \frac{6x^{4}}{(x^{2} + 3)^{2}} + \frac{9x^{2}}{(x^{2} + 3)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-4x^{3}}{(x^{2} + 3)^{2}} + \frac{4x}{(x^{2} + 3)} - \frac{6x^{4}}{(x^{2} + 3)^{2}} + \frac{9x^{2}}{(x^{2} + 3)}\right)}{dx}\\=&-4(\frac{-2(2x + 0)}{(x^{2} + 3)^{3}})x^{3} - \frac{4*3x^{2}}{(x^{2} + 3)^{2}} + 4(\frac{-(2x + 0)}{(x^{2} + 3)^{2}})x + \frac{4}{(x^{2} + 3)} - 6(\frac{-2(2x + 0)}{(x^{2} + 3)^{3}})x^{4} - \frac{6*4x^{3}}{(x^{2} + 3)^{2}} + 9(\frac{-(2x + 0)}{(x^{2} + 3)^{2}})x^{2} + \frac{9*2x}{(x^{2} + 3)}\\=&\frac{16x^{4}}{(x^{2} + 3)^{3}} - \frac{20x^{2}}{(x^{2} + 3)^{2}} + \frac{24x^{5}}{(x^{2} + 3)^{3}} - \frac{42x^{3}}{(x^{2} + 3)^{2}} + \frac{18x}{(x^{2} + 3)} + \frac{4}{(x^{2} + 3)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{16x^{4}}{(x^{2} + 3)^{3}} - \frac{20x^{2}}{(x^{2} + 3)^{2}} + \frac{24x^{5}}{(x^{2} + 3)^{3}} - \frac{42x^{3}}{(x^{2} + 3)^{2}} + \frac{18x}{(x^{2} + 3)} + \frac{4}{(x^{2} + 3)}\right)}{dx}\\=&16(\frac{-3(2x + 0)}{(x^{2} + 3)^{4}})x^{4} + \frac{16*4x^{3}}{(x^{2} + 3)^{3}} - 20(\frac{-2(2x + 0)}{(x^{2} + 3)^{3}})x^{2} - \frac{20*2x}{(x^{2} + 3)^{2}} + 24(\frac{-3(2x + 0)}{(x^{2} + 3)^{4}})x^{5} + \frac{24*5x^{4}}{(x^{2} + 3)^{3}} - 42(\frac{-2(2x + 0)}{(x^{2} + 3)^{3}})x^{3} - \frac{42*3x^{2}}{(x^{2} + 3)^{2}} + 18(\frac{-(2x + 0)}{(x^{2} + 3)^{2}})x + \frac{18}{(x^{2} + 3)} + 4(\frac{-(2x + 0)}{(x^{2} + 3)^{2}})\\=&\frac{-96x^{5}}{(x^{2} + 3)^{4}} + \frac{144x^{3}}{(x^{2} + 3)^{3}} - \frac{48x}{(x^{2} + 3)^{2}} - \frac{144x^{6}}{(x^{2} + 3)^{4}} + \frac{288x^{4}}{(x^{2} + 3)^{3}} - \frac{162x^{2}}{(x^{2} + 3)^{2}} + \frac{18}{(x^{2} + 3)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-96x^{5}}{(x^{2} + 3)^{4}} + \frac{144x^{3}}{(x^{2} + 3)^{3}} - \frac{48x}{(x^{2} + 3)^{2}} - \frac{144x^{6}}{(x^{2} + 3)^{4}} + \frac{288x^{4}}{(x^{2} + 3)^{3}} - \frac{162x^{2}}{(x^{2} + 3)^{2}} + \frac{18}{(x^{2} + 3)}\right)}{dx}\\=&-96(\frac{-4(2x + 0)}{(x^{2} + 3)^{5}})x^{5} - \frac{96*5x^{4}}{(x^{2} + 3)^{4}} + 144(\frac{-3(2x + 0)}{(x^{2} + 3)^{4}})x^{3} + \frac{144*3x^{2}}{(x^{2} + 3)^{3}} - 48(\frac{-2(2x + 0)}{(x^{2} + 3)^{3}})x - \frac{48}{(x^{2} + 3)^{2}} - 144(\frac{-4(2x + 0)}{(x^{2} + 3)^{5}})x^{6} - \frac{144*6x^{5}}{(x^{2} + 3)^{4}} + 288(\frac{-3(2x + 0)}{(x^{2} + 3)^{4}})x^{4} + \frac{288*4x^{3}}{(x^{2} + 3)^{3}} - 162(\frac{-2(2x + 0)}{(x^{2} + 3)^{3}})x^{2} - \frac{162*2x}{(x^{2} + 3)^{2}} + 18(\frac{-(2x + 0)}{(x^{2} + 3)^{2}})\\=&\frac{768x^{6}}{(x^{2} + 3)^{5}} - \frac{1344x^{4}}{(x^{2} + 3)^{4}} + \frac{624x^{2}}{(x^{2} + 3)^{3}} + \frac{1152x^{7}}{(x^{2} + 3)^{5}} - \frac{2592x^{5}}{(x^{2} + 3)^{4}} + \frac{1800x^{3}}{(x^{2} + 3)^{3}} - \frac{360x}{(x^{2} + 3)^{2}} - \frac{48}{(x^{2} + 3)^{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！