There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(1 - 4x)}{(1 + 4x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = - \frac{4x}{(4x + 1)} + \frac{1}{(4x + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( - \frac{4x}{(4x + 1)} + \frac{1}{(4x + 1)}\right)}{dx}\\=& - 4(\frac{-(4 + 0)}{(4x + 1)^{2}})x - \frac{4}{(4x + 1)} + (\frac{-(4 + 0)}{(4x + 1)^{2}})\\=&\frac{16x}{(4x + 1)^{2}} - \frac{4}{(4x + 1)^{2}} - \frac{4}{(4x + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{16x}{(4x + 1)^{2}} - \frac{4}{(4x + 1)^{2}} - \frac{4}{(4x + 1)}\right)}{dx}\\=&16(\frac{-2(4 + 0)}{(4x + 1)^{3}})x + \frac{16}{(4x + 1)^{2}} - 4(\frac{-2(4 + 0)}{(4x + 1)^{3}}) - 4(\frac{-(4 + 0)}{(4x + 1)^{2}})\\=& - \frac{128x}{(4x + 1)^{3}} + \frac{32}{(4x + 1)^{3}} + \frac{32}{(4x + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - \frac{128x}{(4x + 1)^{3}} + \frac{32}{(4x + 1)^{3}} + \frac{32}{(4x + 1)^{2}}\right)}{dx}\\=& - 128(\frac{-3(4 + 0)}{(4x + 1)^{4}})x - \frac{128}{(4x + 1)^{3}} + 32(\frac{-3(4 + 0)}{(4x + 1)^{4}}) + 32(\frac{-2(4 + 0)}{(4x + 1)^{3}})\\=&\frac{1536x}{(4x + 1)^{4}} - \frac{384}{(4x + 1)^{4}} - \frac{384}{(4x + 1)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{1536x}{(4x + 1)^{4}} - \frac{384}{(4x + 1)^{4}} - \frac{384}{(4x + 1)^{3}}\right)}{dx}\\=&1536(\frac{-4(4 + 0)}{(4x + 1)^{5}})x + \frac{1536}{(4x + 1)^{4}} - 384(\frac{-4(4 + 0)}{(4x + 1)^{5}}) - 384(\frac{-3(4 + 0)}{(4x + 1)^{4}})\\=& - \frac{24576x}{(4x + 1)^{5}} + \frac{6144}{(4x + 1)^{5}} + \frac{6144}{(4x + 1)^{4}}\\ \end{split}\end{equation} \]

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