There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ln({x}^{2} + 3x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(x^{2} + 3x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(x^{2} + 3x)\right)}{dx}\\=&\frac{(2x + 3)}{(x^{2} + 3x)}\\=&\frac{2x}{(x^{2} + 3x)} + \frac{3}{(x^{2} + 3x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2x}{(x^{2} + 3x)} + \frac{3}{(x^{2} + 3x)}\right)}{dx}\\=&2(\frac{-(2x + 3)}{(x^{2} + 3x)^{2}})x + \frac{2}{(x^{2} + 3x)} + 3(\frac{-(2x + 3)}{(x^{2} + 3x)^{2}})\\=&\frac{-4x^{2}}{(x^{2} + 3x)^{2}} - \frac{12x}{(x^{2} + 3x)^{2}} + \frac{2}{(x^{2} + 3x)} - \frac{9}{(x^{2} + 3x)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4x^{2}}{(x^{2} + 3x)^{2}} - \frac{12x}{(x^{2} + 3x)^{2}} + \frac{2}{(x^{2} + 3x)} - \frac{9}{(x^{2} + 3x)^{2}}\right)}{dx}\\=&-4(\frac{-2(2x + 3)}{(x^{2} + 3x)^{3}})x^{2} - \frac{4*2x}{(x^{2} + 3x)^{2}} - 12(\frac{-2(2x + 3)}{(x^{2} + 3x)^{3}})x - \frac{12}{(x^{2} + 3x)^{2}} + 2(\frac{-(2x + 3)}{(x^{2} + 3x)^{2}}) - 9(\frac{-2(2x + 3)}{(x^{2} + 3x)^{3}})\\=&\frac{16x^{3}}{(x^{2} + 3x)^{3}} + \frac{72x^{2}}{(x^{2} + 3x)^{3}} - \frac{12x}{(x^{2} + 3x)^{2}} + \frac{108x}{(x^{2} + 3x)^{3}} - \frac{18}{(x^{2} + 3x)^{2}} + \frac{54}{(x^{2} + 3x)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{16x^{3}}{(x^{2} + 3x)^{3}} + \frac{72x^{2}}{(x^{2} + 3x)^{3}} - \frac{12x}{(x^{2} + 3x)^{2}} + \frac{108x}{(x^{2} + 3x)^{3}} - \frac{18}{(x^{2} + 3x)^{2}} + \frac{54}{(x^{2} + 3x)^{3}}\right)}{dx}\\=&16(\frac{-3(2x + 3)}{(x^{2} + 3x)^{4}})x^{3} + \frac{16*3x^{2}}{(x^{2} + 3x)^{3}} + 72(\frac{-3(2x + 3)}{(x^{2} + 3x)^{4}})x^{2} + \frac{72*2x}{(x^{2} + 3x)^{3}} - 12(\frac{-2(2x + 3)}{(x^{2} + 3x)^{3}})x - \frac{12}{(x^{2} + 3x)^{2}} + 108(\frac{-3(2x + 3)}{(x^{2} + 3x)^{4}})x + \frac{108}{(x^{2} + 3x)^{3}} - 18(\frac{-2(2x + 3)}{(x^{2} + 3x)^{3}}) + 54(\frac{-3(2x + 3)}{(x^{2} + 3x)^{4}})\\=&\frac{-96x^{4}}{(x^{2} + 3x)^{4}} - \frac{576x^{3}}{(x^{2} + 3x)^{4}} + \frac{96x^{2}}{(x^{2} + 3x)^{3}} - \frac{1296x^{2}}{(x^{2} + 3x)^{4}} + \frac{288x}{(x^{2} + 3x)^{3}} - \frac{1296x}{(x^{2} + 3x)^{4}} + \frac{216}{(x^{2} + 3x)^{3}} - \frac{12}{(x^{2} + 3x)^{2}} - \frac{486}{(x^{2} + 3x)^{4}}\\ \end{split}\end{equation} \]

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