There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {(x + y)}^{(xy)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (x + y)^{(yx)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (x + y)^{(yx)}\right)}{dx}\\=&((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))\\=&y(x + y)^{(yx)}ln(x + y) + \frac{yx(x + y)^{(yx)}}{(x + y)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( y(x + y)^{(yx)}ln(x + y) + \frac{yx(x + y)^{(yx)}}{(x + y)}\right)}{dx}\\=&y((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))ln(x + y) + \frac{y(x + y)^{(yx)}(1 + 0)}{(x + y)} + (\frac{-(1 + 0)}{(x + y)^{2}})yx(x + y)^{(yx)} + \frac{y(x + y)^{(yx)}}{(x + y)} + \frac{yx((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))}{(x + y)}\\=&y^{2}(x + y)^{(yx)}ln^{2}(x + y) + \frac{2y^{2}x(x + y)^{(yx)}ln(x + y)}{(x + y)} + \frac{2y(x + y)^{(yx)}}{(x + y)} - \frac{yx(x + y)^{(yx)}}{(x + y)^{2}} + \frac{y^{2}x^{2}(x + y)^{(yx)}}{(x + y)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( y^{2}(x + y)^{(yx)}ln^{2}(x + y) + \frac{2y^{2}x(x + y)^{(yx)}ln(x + y)}{(x + y)} + \frac{2y(x + y)^{(yx)}}{(x + y)} - \frac{yx(x + y)^{(yx)}}{(x + y)^{2}} + \frac{y^{2}x^{2}(x + y)^{(yx)}}{(x + y)^{2}}\right)}{dx}\\=&y^{2}((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))ln^{2}(x + y) + \frac{y^{2}(x + y)^{(yx)}*2ln(x + y)(1 + 0)}{(x + y)} + 2(\frac{-(1 + 0)}{(x + y)^{2}})y^{2}x(x + y)^{(yx)}ln(x + y) + \frac{2y^{2}(x + y)^{(yx)}ln(x + y)}{(x + y)} + \frac{2y^{2}x((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))ln(x + y)}{(x + y)} + \frac{2y^{2}x(x + y)^{(yx)}(1 + 0)}{(x + y)(x + y)} + 2(\frac{-(1 + 0)}{(x + y)^{2}})y(x + y)^{(yx)} + \frac{2y((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))}{(x + y)} - (\frac{-2(1 + 0)}{(x + y)^{3}})yx(x + y)^{(yx)} - \frac{y(x + y)^{(yx)}}{(x + y)^{2}} - \frac{yx((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))}{(x + y)^{2}} + (\frac{-2(1 + 0)}{(x + y)^{3}})y^{2}x^{2}(x + y)^{(yx)} + \frac{y^{2}*2x(x + y)^{(yx)}}{(x + y)^{2}} + \frac{y^{2}x^{2}((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))}{(x + y)^{2}}\\=&y^{3}(x + y)^{(yx)}ln^{3}(x + y) + \frac{3y^{3}x(x + y)^{(yx)}ln^{2}(x + y)}{(x + y)} + \frac{6y^{2}(x + y)^{(yx)}ln(x + y)}{(x + y)} - \frac{3y^{2}x(x + y)^{(yx)}ln(x + y)}{(x + y)^{2}} + \frac{3y^{3}x^{2}(x + y)^{(yx)}ln(x + y)}{(x + y)^{2}} + \frac{6y^{2}x(x + y)^{(yx)}}{(x + y)^{2}} - \frac{3y(x + y)^{(yx)}}{(x + y)^{2}} + \frac{2yx(x + y)^{(yx)}}{(x + y)^{3}} - \frac{3y^{2}x^{2}(x + y)^{(yx)}}{(x + y)^{3}} + \frac{y^{3}x^{3}(x + y)^{(yx)}}{(x + y)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( y^{3}(x + y)^{(yx)}ln^{3}(x + y) + \frac{3y^{3}x(x + y)^{(yx)}ln^{2}(x + y)}{(x + y)} + \frac{6y^{2}(x + y)^{(yx)}ln(x + y)}{(x + y)} - \frac{3y^{2}x(x + y)^{(yx)}ln(x + y)}{(x + y)^{2}} + \frac{3y^{3}x^{2}(x + y)^{(yx)}ln(x + y)}{(x + y)^{2}} + \frac{6y^{2}x(x + y)^{(yx)}}{(x + y)^{2}} - \frac{3y(x + y)^{(yx)}}{(x + y)^{2}} + \frac{2yx(x + y)^{(yx)}}{(x + y)^{3}} - \frac{3y^{2}x^{2}(x + y)^{(yx)}}{(x + y)^{3}} + \frac{y^{3}x^{3}(x + y)^{(yx)}}{(x + y)^{3}}\right)}{dx}\\=&y^{3}((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))ln^{3}(x + y) + \frac{y^{3}(x + y)^{(yx)}*3ln^{2}(x + y)(1 + 0)}{(x + y)} + 3(\frac{-(1 + 0)}{(x + y)^{2}})y^{3}x(x + y)^{(yx)}ln^{2}(x + y) + \frac{3y^{3}(x + y)^{(yx)}ln^{2}(x + y)}{(x + y)} + \frac{3y^{3}x((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))ln^{2}(x + y)}{(x + y)} + \frac{3y^{3}x(x + y)^{(yx)}*2ln(x + y)(1 + 0)}{(x + y)(x + y)} + 6(\frac{-(1 + 0)}{(x + y)^{2}})y^{2}(x + y)^{(yx)}ln(x + y) + \frac{6y^{2}((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))ln(x + y)}{(x + y)} + \frac{6y^{2}(x + y)^{(yx)}(1 + 0)}{(x + y)(x + y)} - 3(\frac{-2(1 + 0)}{(x + y)^{3}})y^{2}x(x + y)^{(yx)}ln(x + y) - \frac{3y^{2}(x + y)^{(yx)}ln(x + y)}{(x + y)^{2}} - \frac{3y^{2}x((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))ln(x + y)}{(x + y)^{2}} - \frac{3y^{2}x(x + y)^{(yx)}(1 + 0)}{(x + y)^{2}(x + y)} + 3(\frac{-2(1 + 0)}{(x + y)^{3}})y^{3}x^{2}(x + y)^{(yx)}ln(x + y) + \frac{3y^{3}*2x(x + y)^{(yx)}ln(x + y)}{(x + y)^{2}} + \frac{3y^{3}x^{2}((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))ln(x + y)}{(x + y)^{2}} + \frac{3y^{3}x^{2}(x + y)^{(yx)}(1 + 0)}{(x + y)^{2}(x + y)} + 6(\frac{-2(1 + 0)}{(x + y)^{3}})y^{2}x(x + y)^{(yx)} + \frac{6y^{2}(x + y)^{(yx)}}{(x + y)^{2}} + \frac{6y^{2}x((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))}{(x + y)^{2}} - 3(\frac{-2(1 + 0)}{(x + y)^{3}})y(x + y)^{(yx)} - \frac{3y((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))}{(x + y)^{2}} + 2(\frac{-3(1 + 0)}{(x + y)^{4}})yx(x + y)^{(yx)} + \frac{2y(x + y)^{(yx)}}{(x + y)^{3}} + \frac{2yx((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))}{(x + y)^{3}} - 3(\frac{-3(1 + 0)}{(x + y)^{4}})y^{2}x^{2}(x + y)^{(yx)} - \frac{3y^{2}*2x(x + y)^{(yx)}}{(x + y)^{3}} - \frac{3y^{2}x^{2}((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))}{(x + y)^{3}} + (\frac{-3(1 + 0)}{(x + y)^{4}})y^{3}x^{3}(x + y)^{(yx)} + \frac{y^{3}*3x^{2}(x + y)^{(yx)}}{(x + y)^{3}} + \frac{y^{3}x^{3}((x + y)^{(yx)}((y)ln(x + y) + \frac{(yx)(1 + 0)}{(x + y)}))}{(x + y)^{3}}\\=&y^{4}(x + y)^{(yx)}ln^{4}(x + y) + \frac{4y^{4}x(x + y)^{(yx)}ln^{3}(x + y)}{(x + y)} + \frac{12y^{3}(x + y)^{(yx)}ln^{2}(x + y)}{(x + y)} - \frac{6y^{3}x(x + y)^{(yx)}ln^{2}(x + y)}{(x + y)^{2}} + \frac{6y^{4}x^{2}(x + y)^{(yx)}ln^{2}(x + y)}{(x + y)^{2}} + \frac{24y^{3}x(x + y)^{(yx)}ln(x + y)}{(x + y)^{2}} - \frac{12y^{2}(x + y)^{(yx)}ln(x + y)}{(x + y)^{2}} + \frac{12y^{2}(x + y)^{(yx)}}{(x + y)^{2}} + \frac{8y^{2}x(x + y)^{(yx)}ln(x + y)}{(x + y)^{3}} - \frac{12y^{3}x^{2}(x + y)^{(yx)}ln(x + y)}{(x + y)^{3}} + \frac{4y^{4}x^{3}(x + y)^{(yx)}ln(x + y)}{(x + y)^{3}} - \frac{24y^{2}x(x + y)^{(yx)}}{(x + y)^{3}} + \frac{12y^{3}x^{2}(x + y)^{(yx)}}{(x + y)^{3}} + \frac{8y(x + y)^{(yx)}}{(x + y)^{3}} - \frac{6yx(x + y)^{(yx)}}{(x + y)^{4}} + \frac{11y^{2}x^{2}(x + y)^{(yx)}}{(x + y)^{4}} - \frac{6y^{3}x^{3}(x + y)^{(yx)}}{(x + y)^{4}} + \frac{y^{4}x^{4}(x + y)^{(yx)}}{(x + y)^{4}}\\ \end{split}\end{equation} \]

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