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.
Current location：Derivative function > Derivative function calculation history > Answer
There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ log_{{x}^{x}}^{2{x}^{(3{x}^{4}x)}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = log_{{x}^{x}}^{2{x}^{(3x^{5})}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{{x}^{x}}^{2{x}^{(3x^{5})}}\right)}{dx}\\=&(\frac{(\frac{(2({x}^{(3x^{5})}((3*5x^{4})ln(x) + \frac{(3x^{5})(1)}{(x)})))}{(2{x}^{(3x^{5})})} - \frac{(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{({x}^{x})})}{(ln({x}^{x}))})\\=&\frac{-log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)}{ln({x}^{x})} - \frac{log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{ln({x}^{x})} + \frac{15x^{4}ln(x)}{ln({x}^{x})} + \frac{3x^{4}}{ln({x}^{x})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)}{ln({x}^{x})} - \frac{log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{ln({x}^{x})} + \frac{15x^{4}ln(x)}{ln({x}^{x})} + \frac{3x^{4}}{ln({x}^{x})}\right)}{dx}\\=&\frac{-(\frac{(\frac{(2({x}^{(3x^{5})}((3*5x^{4})ln(x) + \frac{(3x^{5})(1)}{(x)})))}{(2{x}^{(3x^{5})})} - \frac{(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{({x}^{x})})}{(ln({x}^{x}))})ln(x)}{ln({x}^{x})} - \frac{log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{(x)ln({x}^{x})} - \frac{log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)*-({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{2}({x}^{x})({x}^{x})} - \frac{(\frac{(\frac{(2({x}^{(3x^{5})}((3*5x^{4})ln(x) + \frac{(3x^{5})(1)}{(x)})))}{(2{x}^{(3x^{5})})} - \frac{(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{({x}^{x})})}{(ln({x}^{x}))})}{ln({x}^{x})} - \frac{log_{{x}^{x}}^{2{x}^{(3x^{5})}}*-({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{2}({x}^{x})({x}^{x})} + \frac{15*4x^{3}ln(x)}{ln({x}^{x})} + \frac{15x^{4}}{(x)ln({x}^{x})} + \frac{15x^{4}ln(x)*-({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{2}({x}^{x})({x}^{x})} + \frac{3*4x^{3}}{ln({x}^{x})} + \frac{3x^{4}*-({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{2}({x}^{x})({x}^{x})}\\=&\frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln^{2}(x)}{ln^{2}({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)}{ln^{2}({x}^{x})} - \frac{30x^{4}ln^{2}(x)}{ln^{2}({x}^{x})} - \frac{18x^{4}ln(x)}{ln^{2}({x}^{x})} - \frac{log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{xln({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)}{ln^{2}({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{ln^{2}({x}^{x})} - \frac{18x^{4}ln(x)}{ln^{2}({x}^{x})} + \frac{60x^{3}ln(x)}{ln({x}^{x})} - \frac{6x^{4}}{ln^{2}({x}^{x})} + \frac{27x^{3}}{ln({x}^{x})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln^{2}(x)}{ln^{2}({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)}{ln^{2}({x}^{x})} - \frac{30x^{4}ln^{2}(x)}{ln^{2}({x}^{x})} - \frac{18x^{4}ln(x)}{ln^{2}({x}^{x})} - \frac{log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{xln({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)}{ln^{2}({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{ln^{2}({x}^{x})} - \frac{18x^{4}ln(x)}{ln^{2}({x}^{x})} + \frac{60x^{3}ln(x)}{ln({x}^{x})} - \frac{6x^{4}}{ln^{2}({x}^{x})} + \frac{27x^{3}}{ln({x}^{x})}\right)}{dx}\\=&\frac{2(\frac{(\frac{(2({x}^{(3x^{5})}((3*5x^{4})ln(x) + \frac{(3x^{5})(1)}{(x)})))}{(2{x}^{(3x^{5})})} - \frac{(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{({x}^{x})})}{(ln({x}^{x}))})ln^{2}(x)}{ln^{2}({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}*2ln(x)}{(x)ln^{2}({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln^{2}(x)*-2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{3}({x}^{x})({x}^{x})} + \frac{2(\frac{(\frac{(2({x}^{(3x^{5})}((3*5x^{4})ln(x) + \frac{(3x^{5})(1)}{(x)})))}{(2{x}^{(3x^{5})})} - \frac{(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{({x}^{x})})}{(ln({x}^{x}))})ln(x)}{ln^{2}({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}*-2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x)}{ln^{3}({x}^{x})({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{ln^{2}({x}^{x})(x)} - \frac{30*4x^{3}ln^{2}(x)}{ln^{2}({x}^{x})} - \frac{30x^{4}*2ln(x)}{(x)ln^{2}({x}^{x})} - \frac{30x^{4}ln^{2}(x)*-2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{3}({x}^{x})({x}^{x})} - \frac{18*4x^{3}ln(x)}{ln^{2}({x}^{x})} - \frac{18x^{4}*-2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x)}{ln^{3}({x}^{x})({x}^{x})} - \frac{18x^{4}}{ln^{2}({x}^{x})(x)} - \frac{-log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{x^{2}ln({x}^{x})} - \frac{(\frac{(\frac{(2({x}^{(3x^{5})}((3*5x^{4})ln(x) + \frac{(3x^{5})(1)}{(x)})))}{(2{x}^{(3x^{5})})} - \frac{(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{({x}^{x})})}{(ln({x}^{x}))})}{xln({x}^{x})} - \frac{log_{{x}^{x}}^{2{x}^{(3x^{5})}}*-({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{xln^{2}({x}^{x})({x}^{x})} + \frac{2(\frac{(\frac{(2({x}^{(3x^{5})}((3*5x^{4})ln(x) + \frac{(3x^{5})(1)}{(x)})))}{(2{x}^{(3x^{5})})} - \frac{(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{({x}^{x})})}{(ln({x}^{x}))})ln(x)}{ln^{2}({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{(x)ln^{2}({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)*-2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{3}({x}^{x})({x}^{x})} + \frac{2(\frac{(\frac{(2({x}^{(3x^{5})}((3*5x^{4})ln(x) + \frac{(3x^{5})(1)}{(x)})))}{(2{x}^{(3x^{5})})} - \frac{(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{({x}^{x})})}{(ln({x}^{x}))})}{ln^{2}({x}^{x})} + \frac{2log_{{x}^{x}}^{2{x}^{(3x^{5})}}*-2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{3}({x}^{x})({x}^{x})} - \frac{18*4x^{3}ln(x)}{ln^{2}({x}^{x})} - \frac{18x^{4}}{(x)ln^{2}({x}^{x})} - \frac{18x^{4}ln(x)*-2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{3}({x}^{x})({x}^{x})} + \frac{60*3x^{2}ln(x)}{ln({x}^{x})} + \frac{60x^{3}}{(x)ln({x}^{x})} + \frac{60x^{3}ln(x)*-({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{2}({x}^{x})({x}^{x})} - \frac{6*4x^{3}}{ln^{2}({x}^{x})} - \frac{6x^{4}*-2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{3}({x}^{x})({x}^{x})} + \frac{27*3x^{2}}{ln({x}^{x})} + \frac{27x^{3}*-({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{ln^{2}({x}^{x})({x}^{x})}\\=&\frac{-6log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln^{3}(x)}{ln^{3}({x}^{x})} - \frac{12log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln^{2}(x)}{ln^{3}({x}^{x})} + \frac{90x^{4}ln^{3}(x)}{ln^{3}({x}^{x})} + \frac{132x^{4}ln^{2}(x)}{ln^{3}({x}^{x})} + \frac{6log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)}{xln^{2}({x}^{x})} - \frac{6log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln^{2}(x)}{ln^{3}({x}^{x})} - \frac{12log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)}{ln^{3}({x}^{x})} + \frac{66x^{4}ln^{2}(x)}{ln^{3}({x}^{x})} + \frac{84x^{4}ln(x)}{ln^{3}({x}^{x})} + \frac{6log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{xln^{2}({x}^{x})} - \frac{180x^{3}ln^{2}(x)}{ln^{2}({x}^{x})} - \frac{174x^{3}ln(x)}{ln^{2}({x}^{x})} - \frac{132x^{3}ln(x)}{ln^{2}({x}^{x})} + \frac{180x^{2}ln(x)}{ln({x}^{x})} + \frac{log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{x^{2}ln({x}^{x})} + \frac{42x^{4}ln(x)}{ln^{3}({x}^{x})} - \frac{6log_{{x}^{x}}^{2{x}^{(3x^{5})}}ln(x)}{ln^{3}({x}^{x})} - \frac{6log_{{x}^{x}}^{2{x}^{(3x^{5})}}}{ln^{3}({x}^{x})} + \frac{18x^{4}}{ln^{3}({x}^{x})} - \frac{90x^{3}}{ln^{2}({x}^{x})} + \frac{141x^{2}}{ln({x}^{x})}\\ \end{split}\end{equation} \]