There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{-3}{2}cos(2x + 1 + 8lg(2x + 1))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{-3}{2}cos(2x + 8lg(2x + 1) + 1)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{-3}{2}cos(2x + 8lg(2x + 1) + 1)\right)}{dx}\\=&\frac{-3}{2}*-sin(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)\\=&3sin(2x + 8lg(2x + 1) + 1) + \frac{24sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 3sin(2x + 8lg(2x + 1) + 1) + \frac{24sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)ln{10}}\right)}{dx}\\=&3cos(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0) + \frac{24(\frac{-(2 + 0)}{(2x + 1)^{2}})sin(2x + 8lg(2x + 1) + 1)}{ln{10}} + \frac{24*-0sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)ln^{2}{10}} + \frac{24cos(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)}{(2x + 1)ln{10}}\\=&6cos(2x + 8lg(2x + 1) + 1) + \frac{96cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)ln{10}} - \frac{48sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln{10}} + \frac{384cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 6cos(2x + 8lg(2x + 1) + 1) + \frac{96cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)ln{10}} - \frac{48sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln{10}} + \frac{384cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln^{2}{10}}\right)}{dx}\\=&6*-sin(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0) + \frac{96(\frac{-(2 + 0)}{(2x + 1)^{2}})cos(2x + 8lg(2x + 1) + 1)}{ln{10}} + \frac{96*-0cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)ln^{2}{10}} + \frac{96*-sin(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)}{(2x + 1)ln{10}} - \frac{48(\frac{-2(2 + 0)}{(2x + 1)^{3}})sin(2x + 8lg(2x + 1) + 1)}{ln{10}} - \frac{48*-0sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln^{2}{10}} - \frac{48cos(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)}{(2x + 1)^{2}ln{10}} + \frac{384(\frac{-2(2 + 0)}{(2x + 1)^{3}})cos(2x + 8lg(2x + 1) + 1)}{ln^{2}{10}} + \frac{384*-2*0cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln^{3}{10}} + \frac{384*-sin(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)}{(2x + 1)^{2}ln^{2}{10}}\\=&-12sin(2x + 8lg(2x + 1) + 1) - \frac{288sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)ln{10}} - \frac{288cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln{10}} - \frac{2304sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln^{2}{10}} + \frac{192sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln{10}} - \frac{2304cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln^{2}{10}} - \frac{6144sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln^{3}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -12sin(2x + 8lg(2x + 1) + 1) - \frac{288sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)ln{10}} - \frac{288cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln{10}} - \frac{2304sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln^{2}{10}} + \frac{192sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln{10}} - \frac{2304cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln^{2}{10}} - \frac{6144sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln^{3}{10}}\right)}{dx}\\=&-12cos(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0) - \frac{288(\frac{-(2 + 0)}{(2x + 1)^{2}})sin(2x + 8lg(2x + 1) + 1)}{ln{10}} - \frac{288*-0sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)ln^{2}{10}} - \frac{288cos(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)}{(2x + 1)ln{10}} - \frac{288(\frac{-2(2 + 0)}{(2x + 1)^{3}})cos(2x + 8lg(2x + 1) + 1)}{ln{10}} - \frac{288*-0cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln^{2}{10}} - \frac{288*-sin(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)}{(2x + 1)^{2}ln{10}} - \frac{2304(\frac{-2(2 + 0)}{(2x + 1)^{3}})sin(2x + 8lg(2x + 1) + 1)}{ln^{2}{10}} - \frac{2304*-2*0sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln^{3}{10}} - \frac{2304cos(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)}{(2x + 1)^{2}ln^{2}{10}} + \frac{192(\frac{-3(2 + 0)}{(2x + 1)^{4}})sin(2x + 8lg(2x + 1) + 1)}{ln{10}} + \frac{192*-0sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln^{2}{10}} + \frac{192cos(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)}{(2x + 1)^{3}ln{10}} - \frac{2304(\frac{-3(2 + 0)}{(2x + 1)^{4}})cos(2x + 8lg(2x + 1) + 1)}{ln^{2}{10}} - \frac{2304*-2*0cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln^{3}{10}} - \frac{2304*-sin(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)}{(2x + 1)^{3}ln^{2}{10}} - \frac{6144(\frac{-3(2 + 0)}{(2x + 1)^{4}})sin(2x + 8lg(2x + 1) + 1)}{ln^{3}{10}} - \frac{6144*-3*0sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln^{4}{10}} - \frac{6144cos(2x + 8lg(2x + 1) + 1)(2 + \frac{8(2 + 0)}{ln{10}(2x + 1)} + 0)}{(2x + 1)^{3}ln^{3}{10}}\\=&-24cos(2x + 8lg(2x + 1) + 1) - \frac{768cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)ln{10}} + \frac{1152sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln{10}} - \frac{9216cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{2}ln^{2}{10}} + \frac{1536cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln{10}} + \frac{18432sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln^{2}{10}} - \frac{49152cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{3}ln^{3}{10}} - \frac{1152sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{4}ln{10}} + \frac{16896cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{4}ln^{2}{10}} + \frac{73728sin(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{4}ln^{3}{10}} - \frac{98304cos(2x + 8lg(2x + 1) + 1)}{(2x + 1)^{4}ln^{4}{10}}\\ \end{split}\end{equation} \]

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