There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg({x}^{x}sqrt({4}^{{x}^{4}}))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = lg({x}^{x}sqrt({4}^{x^{4}}))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg({x}^{x}sqrt({4}^{x^{4}}))\right)}{dx}\\=&\frac{(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))sqrt({4}^{x^{4}}) + \frac{{x}^{x}({4}^{x^{4}}((4x^{3})ln(4) + \frac{(x^{4})(0)}{(4)}))*\frac{1}{2}}{({4}^{x^{4}})^{\frac{1}{2}}})}{ln{10}({x}^{x}sqrt({4}^{x^{4}}))}\\=&\frac{ln(x)}{ln{10}} + \frac{1}{ln{10}} + \frac{2x^{3}{4}^{(\frac{1}{2}x^{4})}ln(4)}{ln{10}sqrt({4}^{x^{4}})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{ln(x)}{ln{10}} + \frac{1}{ln{10}} + \frac{2x^{3}{4}^{(\frac{1}{2}x^{4})}ln(4)}{ln{10}sqrt({4}^{x^{4}})}\right)}{dx}\\=&\frac{1}{(x)ln{10}} + \frac{ln(x)*-0}{ln^{2}{10}} + \frac{-0}{ln^{2}{10}} + \frac{2*3x^{2}{4}^{(\frac{1}{2}x^{4})}ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{2x^{3}({4}^{(\frac{1}{2}x^{4})}((\frac{1}{2}*4x^{3})ln(4) + \frac{(\frac{1}{2}x^{4})(0)}{(4)}))ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{2x^{3}{4}^{(\frac{1}{2}x^{4})}*0}{(4)ln{10}sqrt({4}^{x^{4}})} + \frac{2x^{3}{4}^{(\frac{1}{2}x^{4})}ln(4)*-0}{ln^{2}{10}sqrt({4}^{x^{4}})} + \frac{2x^{3}{4}^{(\frac{1}{2}x^{4})}ln(4)*-({4}^{x^{4}}((4x^{3})ln(4) + \frac{(x^{4})(0)}{(4)}))*\frac{1}{2}}{ln{10}({4}^{x^{4}})({4}^{x^{4}})^{\frac{1}{2}}}\\=&\frac{-4x^{6}ln^{2}(4)}{ln{10}} + \frac{6x^{2}{4}^{(\frac{1}{2}x^{4})}ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{4x^{6}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{1}{xln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4x^{6}ln^{2}(4)}{ln{10}} + \frac{6x^{2}{4}^{(\frac{1}{2}x^{4})}ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{4x^{6}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{1}{xln{10}}\right)}{dx}\\=&\frac{-4*6x^{5}ln^{2}(4)}{ln{10}} - \frac{4x^{6}*2ln(4)*0}{(4)ln{10}} - \frac{4x^{6}ln^{2}(4)*-0}{ln^{2}{10}} + \frac{6*2x{4}^{(\frac{1}{2}x^{4})}ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{6x^{2}({4}^{(\frac{1}{2}x^{4})}((\frac{1}{2}*4x^{3})ln(4) + \frac{(\frac{1}{2}x^{4})(0)}{(4)}))ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{6x^{2}{4}^{(\frac{1}{2}x^{4})}*0}{(4)ln{10}sqrt({4}^{x^{4}})} + \frac{6x^{2}{4}^{(\frac{1}{2}x^{4})}ln(4)*-0}{ln^{2}{10}sqrt({4}^{x^{4}})} + \frac{6x^{2}{4}^{(\frac{1}{2}x^{4})}ln(4)*-({4}^{x^{4}}((4x^{3})ln(4) + \frac{(x^{4})(0)}{(4)}))*\frac{1}{2}}{ln{10}({4}^{x^{4}})({4}^{x^{4}})^{\frac{1}{2}}} + \frac{4*6x^{5}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{4x^{6}({4}^{(\frac{1}{2}x^{4})}((\frac{1}{2}*4x^{3})ln(4) + \frac{(\frac{1}{2}x^{4})(0)}{(4)}))ln^{2}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{4x^{6}{4}^{(\frac{1}{2}x^{4})}*2ln(4)*0}{(4)ln{10}sqrt({4}^{x^{4}})} + \frac{4x^{6}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)*-0}{ln^{2}{10}sqrt({4}^{x^{4}})} + \frac{4x^{6}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)*-({4}^{x^{4}}((4x^{3})ln(4) + \frac{(x^{4})(0)}{(4)}))*\frac{1}{2}}{ln{10}({4}^{x^{4}})({4}^{x^{4}})^{\frac{1}{2}}} + \frac{-1}{x^{2}ln{10}} + \frac{-0}{xln^{2}{10}}\\=&\frac{-36x^{5}ln^{2}(4)}{ln{10}} + \frac{12x{4}^{(\frac{1}{2}x^{4})}ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{36x^{5}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{8x^{9}{4}^{(\frac{1}{2}x^{4})}ln^{3}(4)}{ln{10}sqrt({4}^{x^{4}})} - \frac{8x^{9}ln^{3}(4)}{ln{10}} - \frac{1}{x^{2}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-36x^{5}ln^{2}(4)}{ln{10}} + \frac{12x{4}^{(\frac{1}{2}x^{4})}ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{36x^{5}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{8x^{9}{4}^{(\frac{1}{2}x^{4})}ln^{3}(4)}{ln{10}sqrt({4}^{x^{4}})} - \frac{8x^{9}ln^{3}(4)}{ln{10}} - \frac{1}{x^{2}ln{10}}\right)}{dx}\\=&\frac{-36*5x^{4}ln^{2}(4)}{ln{10}} - \frac{36x^{5}*2ln(4)*0}{(4)ln{10}} - \frac{36x^{5}ln^{2}(4)*-0}{ln^{2}{10}} + \frac{12 * {4}^{(\frac{1}{2}x^{4})}ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{12x({4}^{(\frac{1}{2}x^{4})}((\frac{1}{2}*4x^{3})ln(4) + \frac{(\frac{1}{2}x^{4})(0)}{(4)}))ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{12x{4}^{(\frac{1}{2}x^{4})}*0}{(4)ln{10}sqrt({4}^{x^{4}})} + \frac{12x{4}^{(\frac{1}{2}x^{4})}ln(4)*-0}{ln^{2}{10}sqrt({4}^{x^{4}})} + \frac{12x{4}^{(\frac{1}{2}x^{4})}ln(4)*-({4}^{x^{4}}((4x^{3})ln(4) + \frac{(x^{4})(0)}{(4)}))*\frac{1}{2}}{ln{10}({4}^{x^{4}})({4}^{x^{4}})^{\frac{1}{2}}} + \frac{36*5x^{4}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{36x^{5}({4}^{(\frac{1}{2}x^{4})}((\frac{1}{2}*4x^{3})ln(4) + \frac{(\frac{1}{2}x^{4})(0)}{(4)}))ln^{2}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{36x^{5}{4}^{(\frac{1}{2}x^{4})}*2ln(4)*0}{(4)ln{10}sqrt({4}^{x^{4}})} + \frac{36x^{5}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)*-0}{ln^{2}{10}sqrt({4}^{x^{4}})} + \frac{36x^{5}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)*-({4}^{x^{4}}((4x^{3})ln(4) + \frac{(x^{4})(0)}{(4)}))*\frac{1}{2}}{ln{10}({4}^{x^{4}})({4}^{x^{4}})^{\frac{1}{2}}} + \frac{8*9x^{8}{4}^{(\frac{1}{2}x^{4})}ln^{3}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{8x^{9}({4}^{(\frac{1}{2}x^{4})}((\frac{1}{2}*4x^{3})ln(4) + \frac{(\frac{1}{2}x^{4})(0)}{(4)}))ln^{3}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{8x^{9}{4}^{(\frac{1}{2}x^{4})}*3ln^{2}(4)*0}{(4)ln{10}sqrt({4}^{x^{4}})} + \frac{8x^{9}{4}^{(\frac{1}{2}x^{4})}ln^{3}(4)*-0}{ln^{2}{10}sqrt({4}^{x^{4}})} + \frac{8x^{9}{4}^{(\frac{1}{2}x^{4})}ln^{3}(4)*-({4}^{x^{4}}((4x^{3})ln(4) + \frac{(x^{4})(0)}{(4)}))*\frac{1}{2}}{ln{10}({4}^{x^{4}})({4}^{x^{4}})^{\frac{1}{2}}} - \frac{8*9x^{8}ln^{3}(4)}{ln{10}} - \frac{8x^{9}*3ln^{2}(4)*0}{(4)ln{10}} - \frac{8x^{9}ln^{3}(4)*-0}{ln^{2}{10}} - \frac{-2}{x^{3}ln{10}} - \frac{-0}{x^{2}ln^{2}{10}}\\=&\frac{-204x^{4}ln^{2}(4)}{ln{10}} + \frac{12 * {4}^{(\frac{1}{2}x^{4})}ln(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{204x^{4}{4}^{(\frac{1}{2}x^{4})}ln^{2}(4)}{ln{10}sqrt({4}^{x^{4}})} + \frac{144x^{8}{4}^{(\frac{1}{2}x^{4})}ln^{3}(4)}{ln{10}sqrt({4}^{x^{4}})} - \frac{144x^{8}ln^{3}(4)}{ln{10}} + \frac{16x^{12}{4}^{(\frac{1}{2}x^{4})}ln^{4}(4)}{ln{10}sqrt({4}^{x^{4}})} - \frac{16x^{12}ln^{4}(4)}{ln{10}} + \frac{2}{x^{3}ln{10}}\\ \end{split}\end{equation} \]

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