There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ xxxsqrt(x) + xxx + xxsqrt(x) + xx + xsqrt(x) + x + sqrt(x) + 1\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = x^{3}sqrt(x) + x^{2}sqrt(x) + xsqrt(x) + x^{2} + x^{3} + x + sqrt(x) + 1\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( x^{3}sqrt(x) + x^{2}sqrt(x) + xsqrt(x) + x^{2} + x^{3} + x + sqrt(x) + 1\right)}{dx}\\=&3x^{2}sqrt(x) + \frac{x^{3}*\frac{1}{2}}{(x)^{\frac{1}{2}}} + 2xsqrt(x) + \frac{x^{2}*\frac{1}{2}}{(x)^{\frac{1}{2}}} + sqrt(x) + \frac{x*\frac{1}{2}}{(x)^{\frac{1}{2}}} + 2x + 3x^{2} + 1 + \frac{\frac{1}{2}}{(x)^{\frac{1}{2}}} + 0\\=&3x^{2}sqrt(x) + \frac{x^{\frac{5}{2}}}{2} + 2xsqrt(x) + \frac{x^{\frac{3}{2}}}{2} + sqrt(x) + \frac{x^{\frac{1}{2}}}{2} + 2x + 3x^{2} + \frac{1}{2x^{\frac{1}{2}}} + 1\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 3x^{2}sqrt(x) + \frac{x^{\frac{5}{2}}}{2} + 2xsqrt(x) + \frac{x^{\frac{3}{2}}}{2} + sqrt(x) + \frac{x^{\frac{1}{2}}}{2} + 2x + 3x^{2} + \frac{1}{2x^{\frac{1}{2}}} + 1\right)}{dx}\\=&3*2xsqrt(x) + \frac{3x^{2}*\frac{1}{2}}{(x)^{\frac{1}{2}}} + \frac{\frac{5}{2}x^{\frac{3}{2}}}{2} + 2sqrt(x) + \frac{2x*\frac{1}{2}}{(x)^{\frac{1}{2}}} + \frac{\frac{3}{2}x^{\frac{1}{2}}}{2} + \frac{\frac{1}{2}}{(x)^{\frac{1}{2}}} + \frac{\frac{1}{2}}{2x^{\frac{1}{2}}} + 2 + 3*2x + \frac{\frac{-1}{2}}{2x^{\frac{3}{2}}} + 0\\=&6xsqrt(x) + \frac{11x^{\frac{3}{2}}}{4} + 2sqrt(x) + \frac{7x^{\frac{1}{2}}}{4} + \frac{3}{4x^{\frac{1}{2}}} + 6x - \frac{1}{4x^{\frac{3}{2}}} + 2\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 6xsqrt(x) + \frac{11x^{\frac{3}{2}}}{4} + 2sqrt(x) + \frac{7x^{\frac{1}{2}}}{4} + \frac{3}{4x^{\frac{1}{2}}} + 6x - \frac{1}{4x^{\frac{3}{2}}} + 2\right)}{dx}\\=&6sqrt(x) + \frac{6x*\frac{1}{2}}{(x)^{\frac{1}{2}}} + \frac{11*\frac{3}{2}x^{\frac{1}{2}}}{4} + \frac{2*\frac{1}{2}}{(x)^{\frac{1}{2}}} + \frac{7*\frac{1}{2}}{4x^{\frac{1}{2}}} + \frac{3*\frac{-1}{2}}{4x^{\frac{3}{2}}} + 6 - \frac{\frac{-3}{2}}{4x^{\frac{5}{2}}} + 0\\=&6sqrt(x) + \frac{57x^{\frac{1}{2}}}{8} + \frac{15}{8x^{\frac{1}{2}}} - \frac{3}{8x^{\frac{3}{2}}} + \frac{3}{8x^{\frac{5}{2}}} + 6\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 6sqrt(x) + \frac{57x^{\frac{1}{2}}}{8} + \frac{15}{8x^{\frac{1}{2}}} - \frac{3}{8x^{\frac{3}{2}}} + \frac{3}{8x^{\frac{5}{2}}} + 6\right)}{dx}\\=&\frac{6*\frac{1}{2}}{(x)^{\frac{1}{2}}} + \frac{57*\frac{1}{2}}{8x^{\frac{1}{2}}} + \frac{15*\frac{-1}{2}}{8x^{\frac{3}{2}}} - \frac{3*\frac{-3}{2}}{8x^{\frac{5}{2}}} + \frac{3*\frac{-5}{2}}{8x^{\frac{7}{2}}} + 0\\=&\frac{105}{16x^{\frac{1}{2}}} - \frac{15}{16x^{\frac{3}{2}}} + \frac{9}{16x^{\frac{5}{2}}} - \frac{15}{16x^{\frac{7}{2}}}\\ \end{split}\end{equation} \]

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