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 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {a}^{{x}^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {a}^{{x}^{x}}\right)}{dx}\\=&({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))\\=&{x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + {x}^{x}{a}^{{x}^{x}}ln(a)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + {x}^{x}{a}^{{x}^{x}}ln(a)\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a)ln(x) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a)ln(x) + \frac{{x}^{x}{a}^{{x}^{x}}*0ln(x)}{(a)} + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{(x)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}*0}{(a)}\\=&{x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a) + {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x) + \frac{{a}^{{x}^{x}}{x}^{x}ln(a)}{x} + {x}^{x}{a}^{{x}^{x}}ln(x)ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a) + {x}^{x}{a}^{{x}^{x}}ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a) + {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x) + \frac{{a}^{{x}^{x}}{x}^{x}ln(a)}{x} + {x}^{x}{a}^{{x}^{x}}ln(x)ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a) + {x}^{x}{a}^{{x}^{x}}ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(x)ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(x)ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}*2ln(x)ln(a)}{(x)} + \frac{{x}^{x}{a}^{{x}^{x}}ln^{2}(x)*0}{(a)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a)ln(x) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a)ln(x) + \frac{{x}^{x}{a}^{{x}^{x}}*0ln(x)}{(a)} + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{(x)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + {x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(x)ln^{2}(a) + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(x)ln^{2}(a)}{(x)} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)*2ln(a)*0}{(a)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a)ln(x) + {x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a)ln(x) + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0ln(x)}{(a)} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{(x)} + \frac{-{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{2}} + \frac{({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)})){x}^{x}ln(a)}{x} + \frac{{a}^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(a)}{x} + \frac{{a}^{{x}^{x}}{x}^{x}*0}{x(a)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{(x)} + \frac{{x}^{x}{a}^{{x}^{x}}ln(x)*0}{(a)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln^{2}(a) + {x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln^{2}(a) + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{(x)} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln(x)*2ln(a)*0}{(a)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}*0}{(a)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a) + {x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a) + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0}{(a)}\\=&{x}^{x}{a}^{{x}^{x}}ln^{3}(x)ln(a) + 3{x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a) + 3{x}^{(2x)}{a}^{{x}^{x}}ln^{3}(x)ln^{2}(a) + 8{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + \frac{3{x}^{x}{a}^{{x}^{x}}ln(x)ln(a)}{x} + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(x)ln^{3}(a) + 2{x}^{(3x)}{a}^{{x}^{x}}ln^{2}(x)ln^{3}(a) + \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x)}{x} + {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + 4{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x) + 2{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln(x) + \frac{2{a}^{{x}^{x}}{x}^{(2x)}ln^{2}(a)}{x} - \frac{{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{2}} + \frac{2{a}^{{x}^{x}}{x}^{x}ln(a)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x} + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{x} + 2{x}^{x}{a}^{{x}^{x}}ln(x)ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln^{2}(x) + 5{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a) + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln^{2}(x) + {x}^{(3x)}{a}^{{x}^{x}}ln(x)ln^{3}(a) + {x}^{x}{a}^{{x}^{x}}ln(a) + 3{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a) + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( {x}^{x}{a}^{{x}^{x}}ln^{3}(x)ln(a) + 3{x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a) + 3{x}^{(2x)}{a}^{{x}^{x}}ln^{3}(x)ln^{2}(a) + 8{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + \frac{3{x}^{x}{a}^{{x}^{x}}ln(x)ln(a)}{x} + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(x)ln^{3}(a) + 2{x}^{(3x)}{a}^{{x}^{x}}ln^{2}(x)ln^{3}(a) + \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x)}{x} + {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + 4{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x) + 2{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln(x) + \frac{2{a}^{{x}^{x}}{x}^{(2x)}ln^{2}(a)}{x} - \frac{{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{2}} + \frac{2{a}^{{x}^{x}}{x}^{x}ln(a)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x} + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{x} + 2{x}^{x}{a}^{{x}^{x}}ln(x)ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln^{2}(x) + 5{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a) + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln^{2}(x) + {x}^{(3x)}{a}^{{x}^{x}}ln(x)ln^{3}(a) + {x}^{x}{a}^{{x}^{x}}ln(a) + 3{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a) + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(x)ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(x)ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}*3ln^{2}(x)ln(a)}{(x)} + \frac{{x}^{x}{a}^{{x}^{x}}ln^{3}(x)*0}{(a)} + 3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(x)ln(a) + 3{x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(x)ln(a) + \frac{3{x}^{x}{a}^{{x}^{x}}*2ln(x)ln(a)}{(x)} + \frac{3{x}^{x}{a}^{{x}^{x}}ln^{2}(x)*0}{(a)} + 3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(x)ln^{2}(a) + 3{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(x)ln^{2}(a) + \frac{3{x}^{(2x)}{a}^{{x}^{x}}*3ln^{2}(x)ln^{2}(a)}{(x)} + \frac{3{x}^{(2x)}{a}^{{x}^{x}}ln^{3}(x)*2ln(a)*0}{(a)} + 8({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + 8{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(x)ln^{2}(a) + \frac{8{x}^{(2x)}{a}^{{x}^{x}}*2ln(x)ln^{2}(a)}{(x)} + \frac{8{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)*2ln(a)*0}{(a)} + \frac{3*-{x}^{x}{a}^{{x}^{x}}ln(x)ln(a)}{x^{2}} + \frac{3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln(a)}{x} + \frac{3{x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln(a)}{x} + \frac{3{x}^{x}{a}^{{x}^{x}}ln(a)}{x(x)} + \frac{3{x}^{x}{a}^{{x}^{x}}ln(x)*0}{x(a)} + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(x)ln^{3}(a) + {x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(x)ln^{3}(a) + \frac{{x}^{(3x)}{a}^{{x}^{x}}*3ln^{2}(x)ln^{3}(a)}{(x)} + \frac{{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(x)*3ln^{2}(a)*0}{(a)} + 2({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(x)ln^{3}(a) + 2{x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(x)ln^{3}(a) + \frac{2{x}^{(3x)}{a}^{{x}^{x}}*2ln(x)ln^{3}(a)}{(x)} + \frac{2{x}^{(3x)}{a}^{{x}^{x}}ln^{2}(x)*3ln^{2}(a)*0}{(a)} + \frac{2*-{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a)}{x^{2}} + \frac{2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln^{2}(a)}{x} + \frac{2{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln^{2}(a)}{x} + \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x(x)} + \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln(x)*2ln(a)*0}{x(a)} + \frac{-{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x)}{x^{2}} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a)ln(x)}{x} + \frac{{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a)ln(x)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0ln(x)}{x(a)} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x(x)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a)ln(x) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a)ln(x) + \frac{{x}^{x}{a}^{{x}^{x}}*0ln(x)}{(a)} + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{(x)} + 4({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a)ln(x) + 4{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a)ln(x) + \frac{4{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0ln(x)}{(a)} + \frac{4{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{(x)} + 2({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(a)ln(x) + 2{x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(a)ln(x) + \frac{2{x}^{(3x)}{a}^{{x}^{x}}*3ln^{2}(a)*0ln(x)}{(a)} + \frac{2{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)}{(x)} + \frac{2*-{a}^{{x}^{x}}{x}^{(2x)}ln^{2}(a)}{x^{2}} + \frac{2({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)})){x}^{(2x)}ln^{2}(a)}{x} + \frac{2{a}^{{x}^{x}}({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{2}(a)}{x} + \frac{2{a}^{{x}^{x}}{x}^{(2x)}*2ln(a)*0}{x(a)} - \frac{-2{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{3}} - \frac{({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)})){x}^{x}ln(a)}{x^{2}} - \frac{{a}^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(a)}{x^{2}} - \frac{{a}^{{x}^{x}}{x}^{x}*0}{x^{2}(a)} + \frac{2*-{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{2}} + \frac{2({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)})){x}^{x}ln(a)}{x} + \frac{2{a}^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(a)}{x} + \frac{2{a}^{{x}^{x}}{x}^{x}*0}{x(a)} + \frac{-{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x^{2}} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a)}{x} + \frac{{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0}{x(a)} + \frac{-{x}^{x}{a}^{{x}^{x}}ln(a)}{x^{2}} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a)}{x} + \frac{{x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a)}{x} + \frac{{x}^{x}{a}^{{x}^{x}}*0}{x(a)} + 2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln(a) + 2{x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln(a) + \frac{2{x}^{x}{a}^{{x}^{x}}ln(a)}{(x)} + \frac{2{x}^{x}{a}^{{x}^{x}}ln(x)*0}{(a)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a)ln^{2}(x) + {x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a)ln^{2}(x) + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0ln^{2}(x)}{(a)} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)*2ln(x)}{(x)} + 5({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln^{2}(a) + 5{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln^{2}(a) + \frac{5{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{(x)} + \frac{5{x}^{(2x)}{a}^{{x}^{x}}ln(x)*2ln(a)*0}{(a)} + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(a)ln^{2}(x) + {x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(a)ln^{2}(x) + \frac{{x}^{(3x)}{a}^{{x}^{x}}*3ln^{2}(a)*0ln^{2}(x)}{(a)} + \frac{{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)*2ln(x)}{(x)} + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln^{3}(a) + {x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln^{3}(a) + \frac{{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)}{(x)} + \frac{{x}^{(3x)}{a}^{{x}^{x}}ln(x)*3ln^{2}(a)*0}{(a)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}*0}{(a)} + 3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a) + 3{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a) + \frac{3{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0}{(a)} + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(a) + {x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(a) + \frac{{x}^{(3x)}{a}^{{x}^{x}}*3ln^{2}(a)*0}{(a)}\\=&{x}^{x}{a}^{{x}^{x}}ln^{4}(x)ln(a) + 4{x}^{x}{a}^{{x}^{x}}ln^{3}(x)ln(a) + 7{x}^{(2x)}{a}^{{x}^{x}}ln^{4}(x)ln^{2}(a) + 27{x}^{(2x)}{a}^{{x}^{x}}ln^{3}(x)ln^{2}(a) + \frac{6{x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a)}{x} + 6{x}^{(3x)}{a}^{{x}^{x}}ln^{4}(x)ln^{3}(a) + 21{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(x)ln^{3}(a) + \frac{12{x}^{x}{a}^{{x}^{x}}ln(x)ln(a)}{x} + {x}^{(4x)}{a}^{{x}^{x}}ln^{4}(x)ln^{4}(a) + 3{x}^{(4x)}{a}^{{x}^{x}}ln^{3}(x)ln^{4}(a) + \frac{15{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a)}{x} + 6{x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a) + 37{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + 24{x}^{(3x)}{a}^{{x}^{x}}ln^{2}(x)ln^{3}(a) + \frac{28{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a)}{x} - \frac{4{x}^{x}{a}^{{x}^{x}}ln(x)ln(a)}{x^{2}} + \frac{3{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln^{2}(x)}{x} + \frac{3{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln^{2}(x)}{x} + \frac{3{x}^{(3x)}{a}^{{x}^{x}}ln^{2}(x)ln^{3}(a)}{x} + 3{x}^{(4x)}{a}^{{x}^{x}}ln^{2}(x)ln^{4}(a) + \frac{6{x}^{(3x)}{a}^{{x}^{x}}ln(x)ln^{3}(a)}{x} - \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a)}{x^{2}} - \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x)}{x^{2}} + \frac{6{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln(x)}{x} + \frac{8{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x)}{x} + \frac{{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{2}} + \frac{3{a}^{{x}^{x}}{x}^{x}ln(a)}{x} + 11{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x) + 12{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln^{2}(x) + 15{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln(x) + \frac{9{a}^{{x}^{x}}{x}^{(2x)}ln^{2}(a)}{x} + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln^{3}(x) + 3{x}^{(4x)}{a}^{{x}^{x}}ln^{4}(a)ln(x) + \frac{3{a}^{{x}^{x}}{x}^{(3x)}ln^{3}(a)}{x} + \frac{{a}^{{x}^{x}}{x}^{(2x)}ln^{2}(a)}{x^{2}} + \frac{2{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{3}} - \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x^{2}} - \frac{2{x}^{x}{a}^{{x}^{x}}ln(a)}{x^{2}} + \frac{3{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)}{x} + \frac{9{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x} + \frac{3{x}^{x}{a}^{{x}^{x}}ln(a)}{x} + 3{x}^{x}{a}^{{x}^{x}}ln(x)ln(a) + {x}^{(4x)}{a}^{{x}^{x}}ln^{4}(a)ln^{3}(x) + 5{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln^{2}(x) + 3{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln^{3}(x) + 17{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a) + 3{x}^{(4x)}{a}^{{x}^{x}}ln^{4}(a)ln^{2}(x) + {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + 9{x}^{(3x)}{a}^{{x}^{x}}ln(x)ln^{3}(a) + {x}^{(4x)}{a}^{{x}^{x}}ln(x)ln^{4}(a) + 7{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a) + 6{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a) + {x}^{x}{a}^{{x}^{x}}ln(a) + {x}^{(4x)}{a}^{{x}^{x}}ln^{4}(a)\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {a}^{{x}^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {a}^{{x}^{x}}\right)}{dx}\\=&({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))\\=&{x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + {x}^{x}{a}^{{x}^{x}}ln(a)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + {x}^{x}{a}^{{x}^{x}}ln(a)\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a)ln(x) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a)ln(x) + \frac{{x}^{x}{a}^{{x}^{x}}*0ln(x)}{(a)} + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{(x)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}*0}{(a)}\\=&{x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a) + {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x) + \frac{{a}^{{x}^{x}}{x}^{x}ln(a)}{x} + {x}^{x}{a}^{{x}^{x}}ln(x)ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a) + {x}^{x}{a}^{{x}^{x}}ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a) + {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x) + \frac{{a}^{{x}^{x}}{x}^{x}ln(a)}{x} + {x}^{x}{a}^{{x}^{x}}ln(x)ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a) + {x}^{x}{a}^{{x}^{x}}ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(x)ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(x)ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}*2ln(x)ln(a)}{(x)} + \frac{{x}^{x}{a}^{{x}^{x}}ln^{2}(x)*0}{(a)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a)ln(x) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a)ln(x) + \frac{{x}^{x}{a}^{{x}^{x}}*0ln(x)}{(a)} + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{(x)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + {x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(x)ln^{2}(a) + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(x)ln^{2}(a)}{(x)} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)*2ln(a)*0}{(a)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a)ln(x) + {x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a)ln(x) + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0ln(x)}{(a)} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{(x)} + \frac{-{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{2}} + \frac{({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)})){x}^{x}ln(a)}{x} + \frac{{a}^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(a)}{x} + \frac{{a}^{{x}^{x}}{x}^{x}*0}{x(a)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{(x)} + \frac{{x}^{x}{a}^{{x}^{x}}ln(x)*0}{(a)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln^{2}(a) + {x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln^{2}(a) + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{(x)} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln(x)*2ln(a)*0}{(a)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}*0}{(a)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a) + {x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a) + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0}{(a)}\\=&{x}^{x}{a}^{{x}^{x}}ln^{3}(x)ln(a) + 3{x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a) + 3{x}^{(2x)}{a}^{{x}^{x}}ln^{3}(x)ln^{2}(a) + 8{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + \frac{3{x}^{x}{a}^{{x}^{x}}ln(x)ln(a)}{x} + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(x)ln^{3}(a) + 2{x}^{(3x)}{a}^{{x}^{x}}ln^{2}(x)ln^{3}(a) + \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x)}{x} + {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + 4{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x) + 2{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln(x) + \frac{2{a}^{{x}^{x}}{x}^{(2x)}ln^{2}(a)}{x} - \frac{{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{2}} + \frac{2{a}^{{x}^{x}}{x}^{x}ln(a)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x} + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{x} + 2{x}^{x}{a}^{{x}^{x}}ln(x)ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln^{2}(x) + 5{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a) + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln^{2}(x) + {x}^{(3x)}{a}^{{x}^{x}}ln(x)ln^{3}(a) + {x}^{x}{a}^{{x}^{x}}ln(a) + 3{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a) + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( {x}^{x}{a}^{{x}^{x}}ln^{3}(x)ln(a) + 3{x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a) + 3{x}^{(2x)}{a}^{{x}^{x}}ln^{3}(x)ln^{2}(a) + 8{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + \frac{3{x}^{x}{a}^{{x}^{x}}ln(x)ln(a)}{x} + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(x)ln^{3}(a) + 2{x}^{(3x)}{a}^{{x}^{x}}ln^{2}(x)ln^{3}(a) + \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x)}{x} + {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + 4{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x) + 2{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln(x) + \frac{2{a}^{{x}^{x}}{x}^{(2x)}ln^{2}(a)}{x} - \frac{{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{2}} + \frac{2{a}^{{x}^{x}}{x}^{x}ln(a)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x} + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{x} + 2{x}^{x}{a}^{{x}^{x}}ln(x)ln(a) + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln^{2}(x) + 5{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a) + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln^{2}(x) + {x}^{(3x)}{a}^{{x}^{x}}ln(x)ln^{3}(a) + {x}^{x}{a}^{{x}^{x}}ln(a) + 3{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a) + {x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(x)ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(x)ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}*3ln^{2}(x)ln(a)}{(x)} + \frac{{x}^{x}{a}^{{x}^{x}}ln^{3}(x)*0}{(a)} + 3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(x)ln(a) + 3{x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(x)ln(a) + \frac{3{x}^{x}{a}^{{x}^{x}}*2ln(x)ln(a)}{(x)} + \frac{3{x}^{x}{a}^{{x}^{x}}ln^{2}(x)*0}{(a)} + 3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(x)ln^{2}(a) + 3{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(x)ln^{2}(a) + \frac{3{x}^{(2x)}{a}^{{x}^{x}}*3ln^{2}(x)ln^{2}(a)}{(x)} + \frac{3{x}^{(2x)}{a}^{{x}^{x}}ln^{3}(x)*2ln(a)*0}{(a)} + 8({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + 8{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(x)ln^{2}(a) + \frac{8{x}^{(2x)}{a}^{{x}^{x}}*2ln(x)ln^{2}(a)}{(x)} + \frac{8{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)*2ln(a)*0}{(a)} + \frac{3*-{x}^{x}{a}^{{x}^{x}}ln(x)ln(a)}{x^{2}} + \frac{3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln(a)}{x} + \frac{3{x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln(a)}{x} + \frac{3{x}^{x}{a}^{{x}^{x}}ln(a)}{x(x)} + \frac{3{x}^{x}{a}^{{x}^{x}}ln(x)*0}{x(a)} + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(x)ln^{3}(a) + {x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(x)ln^{3}(a) + \frac{{x}^{(3x)}{a}^{{x}^{x}}*3ln^{2}(x)ln^{3}(a)}{(x)} + \frac{{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(x)*3ln^{2}(a)*0}{(a)} + 2({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(x)ln^{3}(a) + 2{x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(x)ln^{3}(a) + \frac{2{x}^{(3x)}{a}^{{x}^{x}}*2ln(x)ln^{3}(a)}{(x)} + \frac{2{x}^{(3x)}{a}^{{x}^{x}}ln^{2}(x)*3ln^{2}(a)*0}{(a)} + \frac{2*-{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a)}{x^{2}} + \frac{2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln^{2}(a)}{x} + \frac{2{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln^{2}(a)}{x} + \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x(x)} + \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln(x)*2ln(a)*0}{x(a)} + \frac{-{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x)}{x^{2}} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a)ln(x)}{x} + \frac{{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a)ln(x)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0ln(x)}{x(a)} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x(x)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a)ln(x) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a)ln(x) + \frac{{x}^{x}{a}^{{x}^{x}}*0ln(x)}{(a)} + \frac{{x}^{x}{a}^{{x}^{x}}ln(a)}{(x)} + 4({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a)ln(x) + 4{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a)ln(x) + \frac{4{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0ln(x)}{(a)} + \frac{4{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{(x)} + 2({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(a)ln(x) + 2{x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(a)ln(x) + \frac{2{x}^{(3x)}{a}^{{x}^{x}}*3ln^{2}(a)*0ln(x)}{(a)} + \frac{2{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)}{(x)} + \frac{2*-{a}^{{x}^{x}}{x}^{(2x)}ln^{2}(a)}{x^{2}} + \frac{2({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)})){x}^{(2x)}ln^{2}(a)}{x} + \frac{2{a}^{{x}^{x}}({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{2}(a)}{x} + \frac{2{a}^{{x}^{x}}{x}^{(2x)}*2ln(a)*0}{x(a)} - \frac{-2{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{3}} - \frac{({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)})){x}^{x}ln(a)}{x^{2}} - \frac{{a}^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(a)}{x^{2}} - \frac{{a}^{{x}^{x}}{x}^{x}*0}{x^{2}(a)} + \frac{2*-{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{2}} + \frac{2({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)})){x}^{x}ln(a)}{x} + \frac{2{a}^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(a)}{x} + \frac{2{a}^{{x}^{x}}{x}^{x}*0}{x(a)} + \frac{-{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x^{2}} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a)}{x} + \frac{{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a)}{x} + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0}{x(a)} + \frac{-{x}^{x}{a}^{{x}^{x}}ln(a)}{x^{2}} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a)}{x} + \frac{{x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a)}{x} + \frac{{x}^{x}{a}^{{x}^{x}}*0}{x(a)} + 2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln(a) + 2{x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln(a) + \frac{2{x}^{x}{a}^{{x}^{x}}ln(a)}{(x)} + \frac{2{x}^{x}{a}^{{x}^{x}}ln(x)*0}{(a)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a)ln^{2}(x) + {x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a)ln^{2}(x) + \frac{{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0ln^{2}(x)}{(a)} + \frac{{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)*2ln(x)}{(x)} + 5({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln^{2}(a) + 5{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln^{2}(a) + \frac{5{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{(x)} + \frac{5{x}^{(2x)}{a}^{{x}^{x}}ln(x)*2ln(a)*0}{(a)} + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(a)ln^{2}(x) + {x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(a)ln^{2}(x) + \frac{{x}^{(3x)}{a}^{{x}^{x}}*3ln^{2}(a)*0ln^{2}(x)}{(a)} + \frac{{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)*2ln(x)}{(x)} + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln(x)ln^{3}(a) + {x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(x)ln^{3}(a) + \frac{{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)}{(x)} + \frac{{x}^{(3x)}{a}^{{x}^{x}}ln(x)*3ln^{2}(a)*0}{(a)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){a}^{{x}^{x}}ln(a) + {x}^{x}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln(a) + \frac{{x}^{x}{a}^{{x}^{x}}*0}{(a)} + 3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){a}^{{x}^{x}}ln^{2}(a) + 3{x}^{(2x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{2}(a) + \frac{3{x}^{(2x)}{a}^{{x}^{x}}*2ln(a)*0}{(a)} + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){a}^{{x}^{x}}ln^{3}(a) + {x}^{(3x)}({a}^{{x}^{x}}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(a) + \frac{({x}^{x})(0)}{(a)}))ln^{3}(a) + \frac{{x}^{(3x)}{a}^{{x}^{x}}*3ln^{2}(a)*0}{(a)}\\=&{x}^{x}{a}^{{x}^{x}}ln^{4}(x)ln(a) + 4{x}^{x}{a}^{{x}^{x}}ln^{3}(x)ln(a) + 7{x}^{(2x)}{a}^{{x}^{x}}ln^{4}(x)ln^{2}(a) + 27{x}^{(2x)}{a}^{{x}^{x}}ln^{3}(x)ln^{2}(a) + \frac{6{x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a)}{x} + 6{x}^{(3x)}{a}^{{x}^{x}}ln^{4}(x)ln^{3}(a) + 21{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(x)ln^{3}(a) + \frac{12{x}^{x}{a}^{{x}^{x}}ln(x)ln(a)}{x} + {x}^{(4x)}{a}^{{x}^{x}}ln^{4}(x)ln^{4}(a) + 3{x}^{(4x)}{a}^{{x}^{x}}ln^{3}(x)ln^{4}(a) + \frac{15{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a)}{x} + 6{x}^{x}{a}^{{x}^{x}}ln^{2}(x)ln(a) + 37{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(x)ln^{2}(a) + 24{x}^{(3x)}{a}^{{x}^{x}}ln^{2}(x)ln^{3}(a) + \frac{28{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a)}{x} - \frac{4{x}^{x}{a}^{{x}^{x}}ln(x)ln(a)}{x^{2}} + \frac{3{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln^{2}(x)}{x} + \frac{3{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln^{2}(x)}{x} + \frac{3{x}^{(3x)}{a}^{{x}^{x}}ln^{2}(x)ln^{3}(a)}{x} + 3{x}^{(4x)}{a}^{{x}^{x}}ln^{2}(x)ln^{4}(a) + \frac{6{x}^{(3x)}{a}^{{x}^{x}}ln(x)ln^{3}(a)}{x} - \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a)}{x^{2}} - \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x)}{x^{2}} + \frac{6{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln(x)}{x} + \frac{8{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x)}{x} + \frac{{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{2}} + \frac{3{a}^{{x}^{x}}{x}^{x}ln(a)}{x} + 11{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln(x) + 12{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln^{2}(x) + 15{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln(x) + \frac{9{a}^{{x}^{x}}{x}^{(2x)}ln^{2}(a)}{x} + {x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln^{3}(x) + 3{x}^{(4x)}{a}^{{x}^{x}}ln^{4}(a)ln(x) + \frac{3{a}^{{x}^{x}}{x}^{(3x)}ln^{3}(a)}{x} + \frac{{a}^{{x}^{x}}{x}^{(2x)}ln^{2}(a)}{x^{2}} + \frac{2{a}^{{x}^{x}}{x}^{x}ln(a)}{x^{3}} - \frac{2{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x^{2}} - \frac{2{x}^{x}{a}^{{x}^{x}}ln(a)}{x^{2}} + \frac{3{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)}{x} + \frac{9{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)}{x} + \frac{3{x}^{x}{a}^{{x}^{x}}ln(a)}{x} + 3{x}^{x}{a}^{{x}^{x}}ln(x)ln(a) + {x}^{(4x)}{a}^{{x}^{x}}ln^{4}(a)ln^{3}(x) + 5{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a)ln^{2}(x) + 3{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a)ln^{3}(x) + 17{x}^{(2x)}{a}^{{x}^{x}}ln(x)ln^{2}(a) + 3{x}^{(4x)}{a}^{{x}^{x}}ln^{4}(a)ln^{2}(x) + {x}^{x}{a}^{{x}^{x}}ln(a)ln(x) + 9{x}^{(3x)}{a}^{{x}^{x}}ln(x)ln^{3}(a) + {x}^{(4x)}{a}^{{x}^{x}}ln(x)ln^{4}(a) + 7{x}^{(2x)}{a}^{{x}^{x}}ln^{2}(a) + 6{x}^{(3x)}{a}^{{x}^{x}}ln^{3}(a) + {x}^{x}{a}^{{x}^{x}}ln(a) + {x}^{(4x)}{a}^{{x}^{x}}ln^{4}(a)\\ \end{split}\end{equation} \]