There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ cth(arccos(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cth(arccos(x))\right)}{dx}\\=&(1 - cth^{2}(arccos(x)))(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{cth^{2}(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cth^{2}(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cth^{2}(arccos(x)) + \frac{2cth(arccos(x))(1 - cth^{2}(arccos(x)))(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&\frac{xcth^{2}(arccos(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{2cth(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2cth^{3}(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{xcth^{2}(arccos(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{2cth(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2cth^{3}(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xcth^{2}(arccos(x)) + \frac{cth^{2}(arccos(x))}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{x*2cth(arccos(x))(1 - cth^{2}(arccos(x)))(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cth(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cth(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{2(1 - cth^{2}(arccos(x)))(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cth^{3}(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cth^{3}(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2*3cth^{2}(arccos(x))(1 - cth^{2}(arccos(x)))(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&\frac{3x^{2}cth^{2}(arccos(x))}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{cth^{2}(arccos(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{2xcth(arccos(x))}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xcth^{3}(arccos(x))}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{4xcth(arccos(x))}{(-x^{2} + 1)^{2}} - \frac{8cth^{2}(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)} + \frac{6cth^{4}(arccos(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)} + \frac{4xcth^{3}(arccos(x))}{(-x^{2} + 1)^{2}} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]

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