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。\[ \begin{equation}\begin{split}【1/1】求函数xxln(5{x}^{2} + 6x) 关于 x 的 4 阶导数:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = x^{2}ln(5x^{2} + 6x)\\&\color{blue}{函数的第 1 阶导数:}\\&\frac{d\left( x^{2}ln(5x^{2} + 6x)\right)}{dx}\\=&2xln(5x^{2} + 6x) + \frac{x^{2}(5*2x + 6)}{(5x^{2} + 6x)}\\=&2xln(5x^{2} + 6x) + \frac{10x^{3}}{(5x^{2} + 6x)} + \frac{6x^{2}}{(5x^{2} + 6x)}\\\\ &\color{blue}{函数的第 2 阶导数:} \\&\frac{d\left( 2xln(5x^{2} + 6x) + \frac{10x^{3}}{(5x^{2} + 6x)} + \frac{6x^{2}}{(5x^{2} + 6x)}\right)}{dx}\\=&2ln(5x^{2} + 6x) + \frac{2x(5*2x + 6)}{(5x^{2} + 6x)} + 10(\frac{-(5*2x + 6)}{(5x^{2} + 6x)^{2}})x^{3} + \frac{10*3x^{2}}{(5x^{2} + 6x)} + 6(\frac{-(5*2x + 6)}{(5x^{2} + 6x)^{2}})x^{2} + \frac{6*2x}{(5x^{2} + 6x)}\\=&2ln(5x^{2} + 6x) + \frac{50x^{2}}{(5x^{2} + 6x)} + \frac{24x}{(5x^{2} + 6x)} - \frac{100x^{4}}{(5x^{2} + 6x)^{2}} - \frac{120x^{3}}{(5x^{2} + 6x)^{2}} - \frac{36x^{2}}{(5x^{2} + 6x)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数:} \\&\frac{d\left( 2ln(5x^{2} + 6x) + \frac{50x^{2}}{(5x^{2} + 6x)} + \frac{24x}{(5x^{2} + 6x)} - \frac{100x^{4}}{(5x^{2} + 6x)^{2}} - \frac{120x^{3}}{(5x^{2} + 6x)^{2}} - \frac{36x^{2}}{(5x^{2} + 6x)^{2}}\right)}{dx}\\=&\frac{2(5*2x + 6)}{(5x^{2} + 6x)} + 50(\frac{-(5*2x + 6)}{(5x^{2} + 6x)^{2}})x^{2} + \frac{50*2x}{(5x^{2} + 6x)} + 24(\frac{-(5*2x + 6)}{(5x^{2} + 6x)^{2}})x + \frac{24}{(5x^{2} + 6x)} - 100(\frac{-2(5*2x + 6)}{(5x^{2} + 6x)^{3}})x^{4} - \frac{100*4x^{3}}{(5x^{2} + 6x)^{2}} - 120(\frac{-2(5*2x + 6)}{(5x^{2} + 6x)^{3}})x^{3} - \frac{120*3x^{2}}{(5x^{2} + 6x)^{2}} - 36(\frac{-2(5*2x + 6)}{(5x^{2} + 6x)^{3}})x^{2} - \frac{36*2x}{(5x^{2} + 6x)^{2}}\\=&\frac{120x}{(5x^{2} + 6x)} - \frac{900x^{3}}{(5x^{2} + 6x)^{2}} - \frac{900x^{2}}{(5x^{2} + 6x)^{2}} - \frac{216x}{(5x^{2} + 6x)^{2}} + \frac{2000x^{5}}{(5x^{2} + 6x)^{3}} + \frac{3600x^{4}}{(5x^{2} + 6x)^{3}} + \frac{2160x^{3}}{(5x^{2} + 6x)^{3}} + \frac{432x^{2}}{(5x^{2} + 6x)^{3}} + \frac{36}{(5x^{2} + 6x)}\\\\ &\color{blue}{函数的第 4 阶导数:} \\&\frac{d\left( \frac{120x}{(5x^{2} + 6x)} - \frac{900x^{3}}{(5x^{2} + 6x)^{2}} - \frac{900x^{2}}{(5x^{2} + 6x)^{2}} - \frac{216x}{(5x^{2} + 6x)^{2}} + \frac{2000x^{5}}{(5x^{2} + 6x)^{3}} + \frac{3600x^{4}}{(5x^{2} + 6x)^{3}} + \frac{2160x^{3}}{(5x^{2} + 6x)^{3}} + \frac{432x^{2}}{(5x^{2} + 6x)^{3}} + \frac{36}{(5x^{2} + 6x)}\right)}{dx}\\=&120(\frac{-(5*2x + 6)}{(5x^{2} + 6x)^{2}})x + \frac{120}{(5x^{2} + 6x)} - 900(\frac{-2(5*2x + 6)}{(5x^{2} + 6x)^{3}})x^{3} - \frac{900*3x^{2}}{(5x^{2} + 6x)^{2}} - 900(\frac{-2(5*2x + 6)}{(5x^{2} + 6x)^{3}})x^{2} - \frac{900*2x}{(5x^{2} + 6x)^{2}} - 216(\frac{-2(5*2x + 6)}{(5x^{2} + 6x)^{3}})x - \frac{216}{(5x^{2} + 6x)^{2}} + 2000(\frac{-3(5*2x + 6)}{(5x^{2} + 6x)^{4}})x^{5} + \frac{2000*5x^{4}}{(5x^{2} + 6x)^{3}} + 3600(\frac{-3(5*2x + 6)}{(5x^{2} + 6x)^{4}})x^{4} + \frac{3600*4x^{3}}{(5x^{2} + 6x)^{3}} + 2160(\frac{-3(5*2x + 6)}{(5x^{2} + 6x)^{4}})x^{3} + \frac{2160*3x^{2}}{(5x^{2} + 6x)^{3}} + 432(\frac{-3(5*2x + 6)}{(5x^{2} + 6x)^{4}})x^{2} + \frac{432*2x}{(5x^{2} + 6x)^{3}} + 36(\frac{-(5*2x + 6)}{(5x^{2} + 6x)^{2}})\\=&\frac{-3900x^{2}}{(5x^{2} + 6x)^{2}} - \frac{2880x}{(5x^{2} + 6x)^{2}} + \frac{28000x^{4}}{(5x^{2} + 6x)^{3}} + \frac{43200x^{3}}{(5x^{2} + 6x)^{3}} + \frac{21600x^{2}}{(5x^{2} + 6x)^{3}} + \frac{3456x}{(5x^{2} + 6x)^{3}} - \frac{60000x^{6}}{(5x^{2} + 6x)^{4}} - \frac{144000x^{5}}{(5x^{2} + 6x)^{4}} - \frac{129600x^{4}}{(5x^{2} + 6x)^{4}} - \frac{51840x^{3}}{(5x^{2} + 6x)^{4}} - \frac{7776x^{2}}{(5x^{2} + 6x)^{4}} - \frac{432}{(5x^{2} + 6x)^{2}} + \frac{120}{(5x^{2} + 6x)}\\ \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!