Derivative function：
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
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There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{lg(x)}{arcsin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{lg(x)}{arcsin(x)}\right)}{dx}\\=&\frac{1}{ln{10}(x)arcsin(x)} + lg(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{1}{xln{10}arcsin(x)} - \frac{lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln{10}arcsin(x)} - \frac{lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)}\right)}{dx}\\=&\frac{-1}{x^{2}ln{10}arcsin(x)} + \frac{-0}{xln^{2}{10}arcsin(x)} + \frac{(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{xln{10}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})lg(x)}{arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln{10}(x)arcsin^{2}(x)} - \frac{lg(x)(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-1}{x^{2}ln{10}arcsin(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{2}(x)} - \frac{xlg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{2}(x)} + \frac{2lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln{10}arcsin(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{2}(x)} - \frac{xlg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{2}(x)} + \frac{2lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)}\right)}{dx}\\=&\frac{--2}{x^{3}ln{10}arcsin(x)} - \frac{-0}{x^{2}ln^{2}{10}arcsin(x)} - \frac{(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{x^{2}ln{10}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{xln{10}arcsin^{2}(x)} - \frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} - \frac{-0}{(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}arcsin^{2}(x)} - \frac{(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xlg(x)}{arcsin^{2}(x)} - \frac{lg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}(x)arcsin^{2}(x)} - \frac{xlg(x)(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{xln{10}arcsin^{2}(x)} - \frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} - \frac{-0}{(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}arcsin^{2}(x)} - \frac{(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln{10}(x)arcsin^{3}(x)} + \frac{2lg(x)(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{2}{x^{3}ln{10}arcsin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} - \frac{3}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}arcsin^{2}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} + \frac{4}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{3}(x)} - \frac{3x^{2}lg(x)}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} - \frac{lg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} + \frac{2xlg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{4xlg(x)}{(-x^{2} + 1)^{2}arcsin^{3}(x)} + \frac{2}{(-x^{2} + 1)xln{10}arcsin^{3}(x)} - \frac{6lg(x)}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}ln{10}arcsin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} - \frac{3}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}arcsin^{2}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} + \frac{4}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{3}(x)} - \frac{3x^{2}lg(x)}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} - \frac{lg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} + \frac{2xlg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{4xlg(x)}{(-x^{2} + 1)^{2}arcsin^{3}(x)} + \frac{2}{(-x^{2} + 1)xln{10}arcsin^{3}(x)} - \frac{6lg(x)}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)}\right)}{dx}\\=&\frac{2*-3}{x^{4}ln{10}arcsin(x)} + \frac{2*-0}{x^{3}ln^{2}{10}arcsin(x)} + \frac{2(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{x^{3}ln{10}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{x^{2}ln{10}arcsin^{2}(x)} + \frac{-2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln{10}arcsin^{2}(x)} + \frac{-0}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}arcsin^{2}(x)} + \frac{(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}} - \frac{3(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})}{ln{10}arcsin^{2}(x)} - \frac{3*-0}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}{10}arcsin^{2}(x)} - \frac{3(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{x^{2}ln{10}arcsin^{2}(x)} + \frac{2*-2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln{10}arcsin^{2}(x)} + \frac{2*-0}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}arcsin^{2}(x)} + \frac{2(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{3}(x)} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{3}(x)} + \frac{4*-1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{3}(x)} + \frac{4*-0}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}arcsin^{3}(x)} + \frac{4(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}xln{10}} - \frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}lg(x)}{arcsin^{2}(x)} - \frac{3*2xlg(x)}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}ln{10}(x)arcsin^{2}(x)} - \frac{3x^{2}lg(x)(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})lg(x)}{arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}(x)arcsin^{2}(x)} - \frac{lg(x)(\frac{-2(1)}{arcsin^{3}(x)((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}}})xlg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{2(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xlg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{2lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{2x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln{10}(x)arcsin^{3}(x)} + \frac{2xlg(x)(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{4(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xlg(x)}{arcsin^{3}(x)} + \frac{4lg(x)}{(-x^{2} + 1)^{2}arcsin^{3}(x)} + \frac{4x}{(-x^{2} + 1)^{2}ln{10}(x)arcsin^{3}(x)} + \frac{4xlg(x)(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} + \frac{2(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})}{xln{10}arcsin^{3}(x)} + \frac{2*-1}{(-x^{2} + 1)x^{2}ln{10}arcsin^{3}(x)} + \frac{2*-0}{(-x^{2} + 1)xln^{2}{10}arcsin^{3}(x)} + \frac{2(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)xln{10}} - \frac{6(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)} - \frac{6(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})lg(x)}{(-x^{2} + 1)arcsin^{4}(x)} - \frac{6}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln{10}(x)arcsin^{4}(x)} - \frac{6lg(x)(\frac{-4(1)}{arcsin^{5}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-6}{x^{4}ln{10}arcsin(x)} - \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln{10}arcsin^{2}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{3}{2}}xln{10}arcsin^{2}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln{10}arcsin^{2}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{3}(x)} - \frac{12x}{(-x^{2} + 1)^{\frac{5}{2}}ln{10}arcsin^{2}(x)} + \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln{10}arcsin^{3}(x)} + \frac{18}{(-x^{2} + 1)^{2}ln{10}arcsin^{3}(x)} - \frac{6}{(-x^{2} + 1)x^{2}ln{10}arcsin^{3}(x)} - \frac{12}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{4}(x)} - \frac{15x^{3}lg(x)}{(-x^{2} + 1)^{\frac{7}{2}}arcsin^{2}(x)} - \frac{9xlg(x)}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} + \frac{6x^{2}lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}arcsin^{3}(x)} + \frac{2lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{24x^{2}lg(x)}{(-x^{2} + 1)^{3}arcsin^{3}(x)} + \frac{6lg(x)}{(-x^{2} + 1)^{2}arcsin^{3}(x)} - \frac{6xlg(x)}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)} - \frac{12xlg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}arcsin^{4}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)xln{10}arcsin^{4}(x)} - \frac{18xlg(x)}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{4}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{3}{2}}xln{10}arcsin^{4}(x)} + \frac{24lg(x)}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{5}(x)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{lg(x)}{arcsin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{lg(x)}{arcsin(x)}\right)}{dx}\\=&\frac{1}{ln{10}(x)arcsin(x)} + lg(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{1}{xln{10}arcsin(x)} - \frac{lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln{10}arcsin(x)} - \frac{lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)}\right)}{dx}\\=&\frac{-1}{x^{2}ln{10}arcsin(x)} + \frac{-0}{xln^{2}{10}arcsin(x)} + \frac{(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{xln{10}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})lg(x)}{arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln{10}(x)arcsin^{2}(x)} - \frac{lg(x)(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-1}{x^{2}ln{10}arcsin(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{2}(x)} - \frac{xlg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{2}(x)} + \frac{2lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln{10}arcsin(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{2}(x)} - \frac{xlg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{2}(x)} + \frac{2lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)}\right)}{dx}\\=&\frac{--2}{x^{3}ln{10}arcsin(x)} - \frac{-0}{x^{2}ln^{2}{10}arcsin(x)} - \frac{(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{x^{2}ln{10}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{xln{10}arcsin^{2}(x)} - \frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} - \frac{-0}{(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}arcsin^{2}(x)} - \frac{(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xlg(x)}{arcsin^{2}(x)} - \frac{lg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}(x)arcsin^{2}(x)} - \frac{xlg(x)(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{xln{10}arcsin^{2}(x)} - \frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} - \frac{-0}{(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}arcsin^{2}(x)} - \frac{(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln{10}(x)arcsin^{3}(x)} + \frac{2lg(x)(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{2}{x^{3}ln{10}arcsin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} - \frac{3}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}arcsin^{2}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} + \frac{4}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{3}(x)} - \frac{3x^{2}lg(x)}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} - \frac{lg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} + \frac{2xlg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{4xlg(x)}{(-x^{2} + 1)^{2}arcsin^{3}(x)} + \frac{2}{(-x^{2} + 1)xln{10}arcsin^{3}(x)} - \frac{6lg(x)}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}ln{10}arcsin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} - \frac{3}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}arcsin^{2}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{2}(x)} + \frac{4}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{3}(x)} - \frac{3x^{2}lg(x)}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} - \frac{lg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} + \frac{2xlg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{4xlg(x)}{(-x^{2} + 1)^{2}arcsin^{3}(x)} + \frac{2}{(-x^{2} + 1)xln{10}arcsin^{3}(x)} - \frac{6lg(x)}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)}\right)}{dx}\\=&\frac{2*-3}{x^{4}ln{10}arcsin(x)} + \frac{2*-0}{x^{3}ln^{2}{10}arcsin(x)} + \frac{2(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{x^{3}ln{10}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{x^{2}ln{10}arcsin^{2}(x)} + \frac{-2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln{10}arcsin^{2}(x)} + \frac{-0}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}arcsin^{2}(x)} + \frac{(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}} - \frac{3(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})}{ln{10}arcsin^{2}(x)} - \frac{3*-0}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}{10}arcsin^{2}(x)} - \frac{3(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{x^{2}ln{10}arcsin^{2}(x)} + \frac{2*-2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln{10}arcsin^{2}(x)} + \frac{2*-0}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}arcsin^{2}(x)} + \frac{2(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{3}(x)} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{3}(x)} + \frac{4*-1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{3}(x)} + \frac{4*-0}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}arcsin^{3}(x)} + \frac{4(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}xln{10}} - \frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}lg(x)}{arcsin^{2}(x)} - \frac{3*2xlg(x)}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}ln{10}(x)arcsin^{2}(x)} - \frac{3x^{2}lg(x)(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})lg(x)}{arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}(x)arcsin^{2}(x)} - \frac{lg(x)(\frac{-2(1)}{arcsin^{3}(x)((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}}})xlg(x)}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{2(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xlg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{2lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{2x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln{10}(x)arcsin^{3}(x)} + \frac{2xlg(x)(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{4(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xlg(x)}{arcsin^{3}(x)} + \frac{4lg(x)}{(-x^{2} + 1)^{2}arcsin^{3}(x)} + \frac{4x}{(-x^{2} + 1)^{2}ln{10}(x)arcsin^{3}(x)} + \frac{4xlg(x)(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} + \frac{2(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})}{xln{10}arcsin^{3}(x)} + \frac{2*-1}{(-x^{2} + 1)x^{2}ln{10}arcsin^{3}(x)} + \frac{2*-0}{(-x^{2} + 1)xln^{2}{10}arcsin^{3}(x)} + \frac{2(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)xln{10}} - \frac{6(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)} - \frac{6(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})lg(x)}{(-x^{2} + 1)arcsin^{4}(x)} - \frac{6}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln{10}(x)arcsin^{4}(x)} - \frac{6lg(x)(\frac{-4(1)}{arcsin^{5}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-6}{x^{4}ln{10}arcsin(x)} - \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln{10}arcsin^{2}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{3}{2}}xln{10}arcsin^{2}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln{10}arcsin^{2}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln{10}arcsin^{3}(x)} - \frac{12x}{(-x^{2} + 1)^{\frac{5}{2}}ln{10}arcsin^{2}(x)} + \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln{10}arcsin^{3}(x)} + \frac{18}{(-x^{2} + 1)^{2}ln{10}arcsin^{3}(x)} - \frac{6}{(-x^{2} + 1)x^{2}ln{10}arcsin^{3}(x)} - \frac{12}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}xln{10}arcsin^{4}(x)} - \frac{15x^{3}lg(x)}{(-x^{2} + 1)^{\frac{7}{2}}arcsin^{2}(x)} - \frac{9xlg(x)}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} + \frac{6x^{2}lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}arcsin^{3}(x)} + \frac{2lg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{24x^{2}lg(x)}{(-x^{2} + 1)^{3}arcsin^{3}(x)} + \frac{6lg(x)}{(-x^{2} + 1)^{2}arcsin^{3}(x)} - \frac{6xlg(x)}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)} - \frac{12xlg(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}arcsin^{4}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)xln{10}arcsin^{4}(x)} - \frac{18xlg(x)}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{4}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{3}{2}}xln{10}arcsin^{4}(x)} + \frac{24lg(x)}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{5}(x)}\\ \end{split}\end{equation} \]