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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {({4}^{{x}^{x}})}^{\frac{1}{3}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {4}^{(\frac{1}{3}{x}^{x})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {4}^{(\frac{1}{3}{x}^{x})}\right)}{dx}\\=&({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))\\=&\frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3}\right)}{dx}\\=&\frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)ln(x)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0ln(x)}{3(4)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3(x)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0}{3(4)}\\=&\frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9}\right)}{dx}\\=&\frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(x)ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*2ln(x)ln(4)}{3(x)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)*0}{3(4)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)ln(x)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0ln(x)}{3(4)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3(x)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(x)ln^{2}(4)}{9(x)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)*2ln(4)*0}{9(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)ln(x)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0ln(x)}{9(4)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9(x)} + \frac{-{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{2}} + \frac{({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)})){x}^{x}ln(4)}{3x} + \frac{{4}^{(\frac{1}{3}{x}^{x})}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(4)}{3x} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}*0}{3x(4)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3(x)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*0}{3(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9(x)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*2ln(4)*0}{9(4)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0}{3(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0}{9(4)}\\=&\frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln(4)}{3} + {x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4) + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{2}(4)}{3} + \frac{8{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{x} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{3}(4)}{27} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{3}(4)}{27} + \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{4{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln(x)}{27} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}ln^{2}(4)}{9x} - \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{2}} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3x} + \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{2}(x)}{9} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{2}(x)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{3}(4)}{27} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{3} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{27}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln(4)}{3} + {x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4) + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{2}(4)}{3} + \frac{8{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{x} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{3}(4)}{27} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{3}(4)}{27} + \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{4{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln(x)}{27} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}ln^{2}(4)}{9x} - \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{2}} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3x} + \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{2}(x)}{9} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{2}(x)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{3}(4)}{27} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{3} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{27}\right)}{dx}\\=&\frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(x)ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(x)ln(4)}{3(x)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)*0}{3(4)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4) + {x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(x)ln(4) + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*2ln(x)ln(4)}{(x)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)*0}{(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{2}(4)}{3} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(x)ln^{2}(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(x)ln^{2}(4)}{3(x)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)*2ln(4)*0}{3(4)} + \frac{8({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{8{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(x)ln^{2}(4)}{9} + \frac{8{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(x)ln^{2}(4)}{9(x)} + \frac{8{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)*2ln(4)*0}{9(4)} + \frac{-{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{x^{2}} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{x} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln(4)}{x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{x(x)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*0}{x(4)} + \frac{({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{3}(4)}{27} + \frac{{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(x)ln^{3}(4)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(x)ln^{3}(4)}{27(x)} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)*3ln^{2}(4)*0}{27(4)} + \frac{2({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{3}(4)}{27} + \frac{2{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(x)ln^{3}(4)}{27} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(x)ln^{3}(4)}{27(x)} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)*3ln^{2}(4)*0}{27(4)} + \frac{2*-{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x^{2}} + \frac{2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x} + \frac{2{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln^{2}(4)}{9x} + \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x(x)} + \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*2ln(4)*0}{9x(4)} + \frac{-{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x^{2}} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)ln(x)}{9x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0ln(x)}{9x(4)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x(x)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)ln(x)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0ln(x)}{3(4)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3(x)} + \frac{4({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{4{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)ln(x)}{9} + \frac{4{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0ln(x)}{9(4)} + \frac{4{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9(x)} + \frac{2({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln(x)}{27} + \frac{2{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(4)ln(x)}{27} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(4)*0ln(x)}{27(4)} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{27(x)} + \frac{2*-{4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}ln^{2}(4)}{9x^{2}} + \frac{2({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)})){x}^{(2x)}ln^{2}(4)}{9x} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{2}(4)}{9x} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}*2ln(4)*0}{9x(4)} - \frac{-2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{3}} - \frac{({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)})){x}^{x}ln(4)}{3x^{2}} - \frac{{4}^{(\frac{1}{3}{x}^{x})}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(4)}{3x^{2}} - \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}*0}{3x^{2}(4)} + \frac{2*-{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{2}} + \frac{2({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)})){x}^{x}ln(4)}{3x} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(4)}{3x} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{x}*0}{3x(4)} + \frac{-{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x^{2}} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)}{9x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0}{9x(4)} + \frac{-{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3x^{2}} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3x} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)}{3x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0}{3x(4)} + \frac{2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{2{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln(4)}{3} + \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3(x)} + \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*0}{3(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{2}(x)}{9} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)ln^{2}(x)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0ln^{2}(x)}{9(4)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)*2ln(x)}{9(x)} + \frac{5({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{5{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln^{2}(4)}{9} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9(x)} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*2ln(4)*0}{9(4)} + \frac{({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{2}(x)}{27} + \frac{{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(4)ln^{2}(x)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(4)*0ln^{2}(x)}{27(4)} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)*2ln(x)}{27(x)} + \frac{({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{3}(4)}{27} + \frac{{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln^{3}(4)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{27(x)} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*3ln^{2}(4)*0}{27(4)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0}{3(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{3} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0}{3(4)} + \frac{({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{27} + \frac{{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(4)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(4)*0}{27(4)}\\=&\frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(x)ln(4)}{3} + \frac{4{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln(4)}{3} + \frac{7{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(x)ln^{2}(4)}{9} + 3{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{2}(4) + \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4)}{x} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(x)ln^{3}(4)}{9} + \frac{7{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{3}(4)}{9} + \frac{4{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{x} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(x)ln^{4}(4)}{81} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{4}(4)}{27} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{3x} + 2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4) + \frac{37{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{8{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{3}(4)}{9} + \frac{28{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x} - \frac{4{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3x^{2}} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{2}(x)}{3x} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{2}(x)}{9x} - \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x^{2}} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{3}(4)}{9x} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{4}(4)}{27} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{3}(4)}{9x} - \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x^{2}} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln(x)}{9x} + \frac{8{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}ln^{2}(4)}{9x^{2}} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{x} + \frac{11{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{4{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{2}(x)}{9} + \frac{5{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln(x)}{9} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}ln^{2}(4)}{x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{3}(x)}{9} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(4)ln(x)}{27} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{(3x)}ln^{3}(4)}{9x} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{2}} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{3}} - \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x^{2}} - \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3x^{2}} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{9x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{x} + {x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4) + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(4)ln^{3}(x)}{81} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{2}(x)}{9} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{3}(x)}{9} + \frac{17{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(4)ln^{2}(x)}{27} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{3}(4)}{3} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{4}(4)}{81} + \frac{7{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{9} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(4)}{81}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {({4}^{{x}^{x}})}^{\frac{1}{3}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {4}^{(\frac{1}{3}{x}^{x})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {4}^{(\frac{1}{3}{x}^{x})}\right)}{dx}\\=&({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))\\=&\frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3}\right)}{dx}\\=&\frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)ln(x)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0ln(x)}{3(4)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3(x)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0}{3(4)}\\=&\frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9}\right)}{dx}\\=&\frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(x)ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*2ln(x)ln(4)}{3(x)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)*0}{3(4)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)ln(x)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0ln(x)}{3(4)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3(x)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(x)ln^{2}(4)}{9(x)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)*2ln(4)*0}{9(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)ln(x)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0ln(x)}{9(4)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9(x)} + \frac{-{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{2}} + \frac{({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)})){x}^{x}ln(4)}{3x} + \frac{{4}^{(\frac{1}{3}{x}^{x})}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(4)}{3x} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}*0}{3x(4)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3(x)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*0}{3(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln^{2}(4)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9(x)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*2ln(4)*0}{9(4)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0}{3(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0}{9(4)}\\=&\frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln(4)}{3} + {x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4) + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{2}(4)}{3} + \frac{8{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{x} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{3}(4)}{27} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{3}(4)}{27} + \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{4{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln(x)}{27} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}ln^{2}(4)}{9x} - \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{2}} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3x} + \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{2}(x)}{9} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{2}(x)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{3}(4)}{27} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{3} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{27}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln(4)}{3} + {x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4) + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{2}(4)}{3} + \frac{8{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{x} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{3}(4)}{27} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{3}(4)}{27} + \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{4{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln(x)}{27} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}ln^{2}(4)}{9x} - \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{2}} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3x} + \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{2}(x)}{9} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{2}(x)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{3}(4)}{27} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{3} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{27}\right)}{dx}\\=&\frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(x)ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(x)ln(4)}{3(x)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)*0}{3(4)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4) + {x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(x)ln(4) + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*2ln(x)ln(4)}{(x)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)*0}{(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{2}(4)}{3} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(x)ln^{2}(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(x)ln^{2}(4)}{3(x)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)*2ln(4)*0}{3(4)} + \frac{8({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{8{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(x)ln^{2}(4)}{9} + \frac{8{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(x)ln^{2}(4)}{9(x)} + \frac{8{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)*2ln(4)*0}{9(4)} + \frac{-{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{x^{2}} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{x} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln(4)}{x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{x(x)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*0}{x(4)} + \frac{({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{3}(4)}{27} + \frac{{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(x)ln^{3}(4)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(x)ln^{3}(4)}{27(x)} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)*3ln^{2}(4)*0}{27(4)} + \frac{2({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{3}(4)}{27} + \frac{2{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(x)ln^{3}(4)}{27} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(x)ln^{3}(4)}{27(x)} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)*3ln^{2}(4)*0}{27(4)} + \frac{2*-{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x^{2}} + \frac{2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x} + \frac{2{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln^{2}(4)}{9x} + \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x(x)} + \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*2ln(4)*0}{9x(4)} + \frac{-{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x^{2}} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)ln(x)}{9x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0ln(x)}{9x(4)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x(x)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)ln(x)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0ln(x)}{3(4)} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3(x)} + \frac{4({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{4{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)ln(x)}{9} + \frac{4{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0ln(x)}{9(4)} + \frac{4{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9(x)} + \frac{2({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln(x)}{27} + \frac{2{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(4)ln(x)}{27} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(4)*0ln(x)}{27(4)} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{27(x)} + \frac{2*-{4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}ln^{2}(4)}{9x^{2}} + \frac{2({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)})){x}^{(2x)}ln^{2}(4)}{9x} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{2}(4)}{9x} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}*2ln(4)*0}{9x(4)} - \frac{-2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{3}} - \frac{({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)})){x}^{x}ln(4)}{3x^{2}} - \frac{{4}^{(\frac{1}{3}{x}^{x})}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(4)}{3x^{2}} - \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}*0}{3x^{2}(4)} + \frac{2*-{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{2}} + \frac{2({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)})){x}^{x}ln(4)}{3x} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(4)}{3x} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{x}*0}{3x(4)} + \frac{-{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x^{2}} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)}{9x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0}{9x(4)} + \frac{-{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3x^{2}} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3x} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)}{3x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0}{3x(4)} + \frac{2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3} + \frac{2{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln(4)}{3} + \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3(x)} + \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*0}{3(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{2}(x)}{9} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)ln^{2}(x)}{9} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0ln^{2}(x)}{9(4)} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)*2ln(x)}{9(x)} + \frac{5({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{5{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln^{2}(4)}{9} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9(x)} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*2ln(4)*0}{9(4)} + \frac{({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{2}(x)}{27} + \frac{{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(4)ln^{2}(x)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(4)*0ln^{2}(x)}{27(4)} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)*2ln(x)}{27(x)} + \frac{({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{3}(4)}{27} + \frac{{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(x)ln^{3}(4)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{27(x)} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)*3ln^{2}(4)*0}{27(4)} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{x}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln(4)}{3} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}*0}{3(4)} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{3} + \frac{{x}^{(2x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{2}(4)}{3} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}*2ln(4)*0}{3(4)} + \frac{({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{27} + \frac{{x}^{(3x)}({4}^{(\frac{1}{3}{x}^{x})}((\frac{1}{3}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})))ln(4) + \frac{(\frac{1}{3}{x}^{x})(0)}{(4)}))ln^{3}(4)}{27} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}*3ln^{2}(4)*0}{27(4)}\\=&\frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(x)ln(4)}{3} + \frac{4{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln(4)}{3} + \frac{7{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(x)ln^{2}(4)}{9} + 3{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{2}(4) + \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4)}{x} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(x)ln^{3}(4)}{9} + \frac{7{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{3}(4)}{9} + \frac{4{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{x} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(x)ln^{4}(4)}{81} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(x)ln^{4}(4)}{27} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{3x} + 2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln(4) + \frac{37{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{2}(4)}{9} + \frac{8{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{3}(4)}{9} + \frac{28{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x} - \frac{4{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4)}{3x^{2}} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{2}(x)}{3x} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{2}(x)}{9x} - \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9x^{2}} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{3}(4)}{9x} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(x)ln^{4}(4)}{27} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{3}(4)}{9x} - \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x^{2}} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln(x)}{9x} + \frac{8{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9x} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}ln^{2}(4)}{9x^{2}} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{x} + \frac{11{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln(x)}{9} + \frac{4{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{2}(x)}{9} + \frac{5{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln(x)}{9} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{(2x)}ln^{2}(4)}{x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{3}(x)}{9} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(4)ln(x)}{27} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{(3x)}ln^{3}(4)}{9x} + \frac{{4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{2}} + \frac{2 * {4}^{(\frac{1}{3}{x}^{x})}{x}^{x}ln(4)}{3x^{3}} - \frac{2{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9x^{2}} - \frac{2{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3x^{2}} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{9x} + \frac{{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{x} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{x} + {x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln(4) + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(4)ln^{3}(x)}{81} + \frac{5{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)ln^{2}(x)}{9} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)ln^{3}(x)}{9} + \frac{17{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{2}(4)}{9} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(4)ln^{2}(x)}{27} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)ln(x)}{3} + \frac{{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{3}(4)}{3} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln(x)ln^{4}(4)}{81} + \frac{7{x}^{(2x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{2}(4)}{9} + \frac{2{x}^{(3x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{3}(4)}{9} + \frac{{x}^{x}{4}^{(\frac{1}{3}{x}^{x})}ln(4)}{3} + \frac{{x}^{(4x)}{4}^{(\frac{1}{3}{x}^{x})}ln^{4}(4)}{81}\\ \end{split}\end{equation} \]