There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(1 + {10}^{(xx)})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = lg({10}^{x^{2}} + 1)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg({10}^{x^{2}} + 1)\right)}{dx}\\=&\frac{(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{ln{10}({10}^{x^{2}} + 1)}\\=&\frac{2x{10}^{x^{2}}ln(10)}{({10}^{x^{2}} + 1)ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2x{10}^{x^{2}}ln(10)}{({10}^{x^{2}} + 1)ln{10}}\right)}{dx}\\=&\frac{2(\frac{-(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{2}})x{10}^{x^{2}}ln(10)}{ln{10}} + \frac{2 * {10}^{x^{2}}ln(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{2x({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)}))ln(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{2x{10}^{x^{2}}*0}{({10}^{x^{2}} + 1)(10)ln{10}} + \frac{2x{10}^{x^{2}}ln(10)*-0}{({10}^{x^{2}} + 1)ln^{2}{10}}\\=&\frac{-4x^{2}{10}^{(2x^{2})}ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} + \frac{2 * {10}^{x^{2}}ln(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{4x^{2}{10}^{x^{2}}ln^{2}(10)}{({10}^{x^{2}} + 1)ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4x^{2}{10}^{(2x^{2})}ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} + \frac{2 * {10}^{x^{2}}ln(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{4x^{2}{10}^{x^{2}}ln^{2}(10)}{({10}^{x^{2}} + 1)ln{10}}\right)}{dx}\\=&\frac{-4(\frac{-2(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{3}})x^{2}{10}^{(2x^{2})}ln^{2}(10)}{ln{10}} - \frac{4*2x{10}^{(2x^{2})}ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{4x^{2}({10}^{(2x^{2})}((2*2x)ln(10) + \frac{(2x^{2})(0)}{(10)}))ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{4x^{2}{10}^{(2x^{2})}*2ln(10)*0}{({10}^{x^{2}} + 1)^{2}(10)ln{10}} - \frac{4x^{2}{10}^{(2x^{2})}ln^{2}(10)*-0}{({10}^{x^{2}} + 1)^{2}ln^{2}{10}} + \frac{2(\frac{-(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{2}}){10}^{x^{2}}ln(10)}{ln{10}} + \frac{2({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)}))ln(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{2 * {10}^{x^{2}}*0}{({10}^{x^{2}} + 1)(10)ln{10}} + \frac{2 * {10}^{x^{2}}ln(10)*-0}{({10}^{x^{2}} + 1)ln^{2}{10}} + \frac{4(\frac{-(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{2}})x^{2}{10}^{x^{2}}ln^{2}(10)}{ln{10}} + \frac{4*2x{10}^{x^{2}}ln^{2}(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{4x^{2}({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)}))ln^{2}(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{4x^{2}{10}^{x^{2}}*2ln(10)*0}{({10}^{x^{2}} + 1)(10)ln{10}} + \frac{4x^{2}{10}^{x^{2}}ln^{2}(10)*-0}{({10}^{x^{2}} + 1)ln^{2}{10}}\\=&\frac{16x^{3}{10}^{(3x^{2})}ln^{3}(10)}{({10}^{x^{2}} + 1)^{3}ln{10}} - \frac{8x{10}^{(2x^{2})}ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{16x^{3}{10}^{(2x^{2})}ln^{3}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{4x{10}^{(2(x^{2}))}ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} + \frac{12x{10}^{x^{2}}ln^{2}(10)}{({10}^{x^{2}} + 1)ln{10}} - \frac{8x^{3}{10}^{(2(x^{2}))}ln^{3}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} + \frac{8x^{3}{10}^{x^{2}}ln^{3}(10)}{({10}^{x^{2}} + 1)ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{16x^{3}{10}^{(3x^{2})}ln^{3}(10)}{({10}^{x^{2}} + 1)^{3}ln{10}} - \frac{8x{10}^{(2x^{2})}ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{16x^{3}{10}^{(2x^{2})}ln^{3}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{4x{10}^{(2(x^{2}))}ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} + \frac{12x{10}^{x^{2}}ln^{2}(10)}{({10}^{x^{2}} + 1)ln{10}} - \frac{8x^{3}{10}^{(2(x^{2}))}ln^{3}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} + \frac{8x^{3}{10}^{x^{2}}ln^{3}(10)}{({10}^{x^{2}} + 1)ln{10}}\right)}{dx}\\=&\frac{16(\frac{-3(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{4}})x^{3}{10}^{(3x^{2})}ln^{3}(10)}{ln{10}} + \frac{16*3x^{2}{10}^{(3x^{2})}ln^{3}(10)}{({10}^{x^{2}} + 1)^{3}ln{10}} + \frac{16x^{3}({10}^{(3x^{2})}((3*2x)ln(10) + \frac{(3x^{2})(0)}{(10)}))ln^{3}(10)}{({10}^{x^{2}} + 1)^{3}ln{10}} + \frac{16x^{3}{10}^{(3x^{2})}*3ln^{2}(10)*0}{({10}^{x^{2}} + 1)^{3}(10)ln{10}} + \frac{16x^{3}{10}^{(3x^{2})}ln^{3}(10)*-0}{({10}^{x^{2}} + 1)^{3}ln^{2}{10}} - \frac{8(\frac{-2(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{3}})x{10}^{(2x^{2})}ln^{2}(10)}{ln{10}} - \frac{8 * {10}^{(2x^{2})}ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{8x({10}^{(2x^{2})}((2*2x)ln(10) + \frac{(2x^{2})(0)}{(10)}))ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{8x{10}^{(2x^{2})}*2ln(10)*0}{({10}^{x^{2}} + 1)^{2}(10)ln{10}} - \frac{8x{10}^{(2x^{2})}ln^{2}(10)*-0}{({10}^{x^{2}} + 1)^{2}ln^{2}{10}} - \frac{16(\frac{-2(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{3}})x^{3}{10}^{(2x^{2})}ln^{3}(10)}{ln{10}} - \frac{16*3x^{2}{10}^{(2x^{2})}ln^{3}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{16x^{3}({10}^{(2x^{2})}((2*2x)ln(10) + \frac{(2x^{2})(0)}{(10)}))ln^{3}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{16x^{3}{10}^{(2x^{2})}*3ln^{2}(10)*0}{({10}^{x^{2}} + 1)^{2}(10)ln{10}} - \frac{16x^{3}{10}^{(2x^{2})}ln^{3}(10)*-0}{({10}^{x^{2}} + 1)^{2}ln^{2}{10}} - \frac{4(\frac{-2(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{3}})x{10}^{(2(x^{2}))}ln^{2}(10)}{ln{10}} - \frac{4 * {10}^{(2(x^{2}))}ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{4x({10}^{(2(x^{2}))}((2(2x))ln(10) + \frac{(2(x^{2}))(0)}{(10)}))ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{4x{10}^{(2(x^{2}))}*2ln(10)*0}{({10}^{x^{2}} + 1)^{2}(10)ln{10}} - \frac{4x{10}^{(2(x^{2}))}ln^{2}(10)*-0}{({10}^{x^{2}} + 1)^{2}ln^{2}{10}} + \frac{12(\frac{-(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{2}})x{10}^{x^{2}}ln^{2}(10)}{ln{10}} + \frac{12 * {10}^{x^{2}}ln^{2}(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{12x({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)}))ln^{2}(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{12x{10}^{x^{2}}*2ln(10)*0}{({10}^{x^{2}} + 1)(10)ln{10}} + \frac{12x{10}^{x^{2}}ln^{2}(10)*-0}{({10}^{x^{2}} + 1)ln^{2}{10}} - \frac{8(\frac{-2(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{3}})x^{3}{10}^{(2(x^{2}))}ln^{3}(10)}{ln{10}} - \frac{8*3x^{2}{10}^{(2(x^{2}))}ln^{3}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{8x^{3}({10}^{(2(x^{2}))}((2(2x))ln(10) + \frac{(2(x^{2}))(0)}{(10)}))ln^{3}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{8x^{3}{10}^{(2(x^{2}))}*3ln^{2}(10)*0}{({10}^{x^{2}} + 1)^{2}(10)ln{10}} - \frac{8x^{3}{10}^{(2(x^{2}))}ln^{3}(10)*-0}{({10}^{x^{2}} + 1)^{2}ln^{2}{10}} + \frac{8(\frac{-(({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)})) + 0)}{({10}^{x^{2}} + 1)^{2}})x^{3}{10}^{x^{2}}ln^{3}(10)}{ln{10}} + \frac{8*3x^{2}{10}^{x^{2}}ln^{3}(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{8x^{3}({10}^{x^{2}}((2x)ln(10) + \frac{(x^{2})(0)}{(10)}))ln^{3}(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{8x^{3}{10}^{x^{2}}*3ln^{2}(10)*0}{({10}^{x^{2}} + 1)(10)ln{10}} + \frac{8x^{3}{10}^{x^{2}}ln^{3}(10)*-0}{({10}^{x^{2}} + 1)ln^{2}{10}}\\=&\frac{-96x^{4}{10}^{(4x^{2})}ln^{4}(10)}{({10}^{x^{2}} + 1)^{4}ln{10}} + \frac{96x^{2}{10}^{(3x^{2})}ln^{3}(10)}{({10}^{x^{2}} + 1)^{3}ln{10}} + \frac{192x^{4}{10}^{(3x^{2})}ln^{4}(10)}{({10}^{x^{2}} + 1)^{3}ln{10}} - \frac{12 * {10}^{(2x^{2})}ln^{2}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{120x^{2}{10}^{(2x^{2})}ln^{3}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{96x^{4}{10}^{(2x^{2})}ln^{4}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} - \frac{24x^{2}{10}^{(2(x^{2}))}ln^{3}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} + \frac{12 * {10}^{x^{2}}ln^{2}(10)}{({10}^{x^{2}} + 1)ln{10}} + \frac{48x^{2}{10}^{x^{2}}ln^{3}(10)}{({10}^{x^{2}} + 1)ln{10}} - \frac{16x^{4}{10}^{(2(x^{2}))}ln^{4}(10)}{({10}^{x^{2}} + 1)^{2}ln{10}} + \frac{16x^{4}{10}^{x^{2}}ln^{4}(10)}{({10}^{x^{2}} + 1)ln{10}}\\ \end{split}\end{equation} \]

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