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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ((4{e}^{x} + 2 * {3}^{2}x){\frac{1}{3}}^{x})dx\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = 4dx{e}^{x}{\frac{1}{3}}^{x} + 18dx^{2}{\frac{1}{3}}^{x}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( 4dx{e}^{x}{\frac{1}{3}}^{x} + 18dx^{2}{\frac{1}{3}}^{x}\right)}{dx}\\=&4d{e}^{x}{\frac{1}{3}}^{x} + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 18d*2x{\frac{1}{3}}^{x} + 18dx^{2}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))\\=&4d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 18dx^{2}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 36dx{\frac{1}{3}}^{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 4d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 18dx^{2}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 36dx{\frac{1}{3}}^{x}\right)}{dx}\\=&4d({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 4d{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 4d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(\frac{1}{3}) + \frac{4dx{\frac{1}{3}}^{x}{e}^{x}*0}{(\frac{1}{3})} + 4d{e}^{x}{\frac{1}{3}}^{x} + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 18d*2x{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 18dx^{2}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{18dx^{2}{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 36d{\frac{1}{3}}^{x} + 36dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))\\=&8d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 8d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 72dx{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 18dx^{2}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 36d{\frac{1}{3}}^{x}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 8d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 8d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 72dx{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 18dx^{2}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 36d{\frac{1}{3}}^{x}\right)}{dx}\\=&8d({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln(\frac{1}{3}) + 8d{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(\frac{1}{3}) + \frac{8d{\frac{1}{3}}^{x}{e}^{x}*0}{(\frac{1}{3})} + 8d({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 8d{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 4d{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 4dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln^{2}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln^{2}(\frac{1}{3}) + \frac{4dx{\frac{1}{3}}^{x}{e}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 4d{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{4dx{e}^{x}{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 4d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(\frac{1}{3}) + \frac{4dx{\frac{1}{3}}^{x}{e}^{x}*0}{(\frac{1}{3})} + 4d{e}^{x}{\frac{1}{3}}^{x} + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 72d{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 72dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{72dx{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 18d*2x{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 18dx^{2}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln^{2}(\frac{1}{3}) + \frac{18dx^{2}{\frac{1}{3}}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 36d({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))\\=&12d{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 12d{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 12d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 12d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln^{3}(\frac{1}{3}) + 8dx{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 8dx{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 108d{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 108dx{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 18dx^{2}{\frac{1}{3}}^{x}ln^{3}(\frac{1}{3})\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 12d{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 12d{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 12d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 12d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln^{3}(\frac{1}{3}) + 8dx{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 8dx{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 108d{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 108dx{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 18dx^{2}{\frac{1}{3}}^{x}ln^{3}(\frac{1}{3})\right)}{dx}\\=&12d({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln^{2}(\frac{1}{3}) + 12d{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln^{2}(\frac{1}{3}) + \frac{12d{\frac{1}{3}}^{x}{e}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 12d({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x}ln(\frac{1}{3}) + 12d{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{12d{e}^{x}{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 12d({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln(\frac{1}{3}) + 12d{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(\frac{1}{3}) + \frac{12d{\frac{1}{3}}^{x}{e}^{x}*0}{(\frac{1}{3})} + 12d({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 12d{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 4d{\frac{1}{3}}^{x}{e}^{x}ln^{3}(\frac{1}{3}) + 4dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln^{3}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln^{3}(\frac{1}{3}) + \frac{4dx{\frac{1}{3}}^{x}{e}^{x}*3ln^{2}(\frac{1}{3})*0}{(\frac{1}{3})} + 8d{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 8dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln^{2}(\frac{1}{3}) + 8dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln^{2}(\frac{1}{3}) + \frac{8dx{\frac{1}{3}}^{x}{e}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 8d{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 8dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x}ln(\frac{1}{3}) + 8dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{8dx{e}^{x}{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 4d{e}^{x}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln^{2}(\frac{1}{3}) + \frac{4dx{e}^{x}{\frac{1}{3}}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 4d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(\frac{1}{3}) + \frac{4dx{\frac{1}{3}}^{x}{e}^{x}*0}{(\frac{1}{3})} + 4d{e}^{x}{\frac{1}{3}}^{x} + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 108d({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{108d{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 108d{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 108dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln^{2}(\frac{1}{3}) + \frac{108dx{\frac{1}{3}}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 18d*2x{\frac{1}{3}}^{x}ln^{3}(\frac{1}{3}) + 18dx^{2}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln^{3}(\frac{1}{3}) + \frac{18dx^{2}{\frac{1}{3}}^{x}*3ln^{2}(\frac{1}{3})*0}{(\frac{1}{3})}\\=&16d{\frac{1}{3}}^{x}{e}^{x}ln^{3}(\frac{1}{3}) + 32d{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 32d{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 16d{e}^{x}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln^{4}(\frac{1}{3}) + 16d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 16d{e}^{x}{\frac{1}{3}}^{x} + 12dx{\frac{1}{3}}^{x}{e}^{x}ln^{3}(\frac{1}{3}) + 12dx{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 12dx{e}^{x}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 12dx{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x}ln^{3}(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 216d{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 144dx{\frac{1}{3}}^{x}ln^{3}(\frac{1}{3}) + 18dx^{2}{\frac{1}{3}}^{x}ln^{4}(\frac{1}{3})\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ((4{e}^{x} + 2 * {3}^{2}x){\frac{1}{3}}^{x})dx\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = 4dx{e}^{x}{\frac{1}{3}}^{x} + 18dx^{2}{\frac{1}{3}}^{x}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( 4dx{e}^{x}{\frac{1}{3}}^{x} + 18dx^{2}{\frac{1}{3}}^{x}\right)}{dx}\\=&4d{e}^{x}{\frac{1}{3}}^{x} + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 18d*2x{\frac{1}{3}}^{x} + 18dx^{2}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))\\=&4d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 18dx^{2}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 36dx{\frac{1}{3}}^{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 4d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 18dx^{2}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 36dx{\frac{1}{3}}^{x}\right)}{dx}\\=&4d({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 4d{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 4d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(\frac{1}{3}) + \frac{4dx{\frac{1}{3}}^{x}{e}^{x}*0}{(\frac{1}{3})} + 4d{e}^{x}{\frac{1}{3}}^{x} + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 18d*2x{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 18dx^{2}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{18dx^{2}{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 36d{\frac{1}{3}}^{x} + 36dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))\\=&8d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 8d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 72dx{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 18dx^{2}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 36d{\frac{1}{3}}^{x}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 8d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 8d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 72dx{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 18dx^{2}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 36d{\frac{1}{3}}^{x}\right)}{dx}\\=&8d({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln(\frac{1}{3}) + 8d{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(\frac{1}{3}) + \frac{8d{\frac{1}{3}}^{x}{e}^{x}*0}{(\frac{1}{3})} + 8d({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 8d{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 4d{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 4dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln^{2}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln^{2}(\frac{1}{3}) + \frac{4dx{\frac{1}{3}}^{x}{e}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 4d{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{4dx{e}^{x}{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 4d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(\frac{1}{3}) + \frac{4dx{\frac{1}{3}}^{x}{e}^{x}*0}{(\frac{1}{3})} + 4d{e}^{x}{\frac{1}{3}}^{x} + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 72d{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 72dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{72dx{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 18d*2x{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 18dx^{2}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln^{2}(\frac{1}{3}) + \frac{18dx^{2}{\frac{1}{3}}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 36d({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))\\=&12d{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 12d{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 12d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 12d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln^{3}(\frac{1}{3}) + 8dx{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 8dx{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 108d{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 108dx{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 18dx^{2}{\frac{1}{3}}^{x}ln^{3}(\frac{1}{3})\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 12d{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 12d{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 12d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 12d{e}^{x}{\frac{1}{3}}^{x} + 4dx{\frac{1}{3}}^{x}{e}^{x}ln^{3}(\frac{1}{3}) + 8dx{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 8dx{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 108d{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 108dx{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 18dx^{2}{\frac{1}{3}}^{x}ln^{3}(\frac{1}{3})\right)}{dx}\\=&12d({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln^{2}(\frac{1}{3}) + 12d{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln^{2}(\frac{1}{3}) + \frac{12d{\frac{1}{3}}^{x}{e}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 12d({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x}ln(\frac{1}{3}) + 12d{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{12d{e}^{x}{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 12d({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln(\frac{1}{3}) + 12d{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(\frac{1}{3}) + \frac{12d{\frac{1}{3}}^{x}{e}^{x}*0}{(\frac{1}{3})} + 12d({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 12d{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 4d{\frac{1}{3}}^{x}{e}^{x}ln^{3}(\frac{1}{3}) + 4dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln^{3}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln^{3}(\frac{1}{3}) + \frac{4dx{\frac{1}{3}}^{x}{e}^{x}*3ln^{2}(\frac{1}{3})*0}{(\frac{1}{3})} + 8d{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 8dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln^{2}(\frac{1}{3}) + 8dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln^{2}(\frac{1}{3}) + \frac{8dx{\frac{1}{3}}^{x}{e}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 8d{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 8dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x}ln(\frac{1}{3}) + 8dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{8dx{e}^{x}{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 4d{e}^{x}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln^{2}(\frac{1}{3}) + \frac{4dx{e}^{x}{\frac{1}{3}}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 4d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 4dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})){e}^{x}ln(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(\frac{1}{3}) + \frac{4dx{\frac{1}{3}}^{x}{e}^{x}*0}{(\frac{1}{3})} + 4d{e}^{x}{\frac{1}{3}}^{x} + 4dx({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})){\frac{1}{3}}^{x} + 4dx{e}^{x}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})})) + 108d({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln(\frac{1}{3}) + \frac{108d{\frac{1}{3}}^{x}*0}{(\frac{1}{3})} + 108d{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 108dx({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln^{2}(\frac{1}{3}) + \frac{108dx{\frac{1}{3}}^{x}*2ln(\frac{1}{3})*0}{(\frac{1}{3})} + 18d*2x{\frac{1}{3}}^{x}ln^{3}(\frac{1}{3}) + 18dx^{2}({\frac{1}{3}}^{x}((1)ln(\frac{1}{3}) + \frac{(x)(0)}{(\frac{1}{3})}))ln^{3}(\frac{1}{3}) + \frac{18dx^{2}{\frac{1}{3}}^{x}*3ln^{2}(\frac{1}{3})*0}{(\frac{1}{3})}\\=&16d{\frac{1}{3}}^{x}{e}^{x}ln^{3}(\frac{1}{3}) + 32d{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 32d{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 16d{e}^{x}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln^{4}(\frac{1}{3}) + 16d{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 16d{e}^{x}{\frac{1}{3}}^{x} + 12dx{\frac{1}{3}}^{x}{e}^{x}ln^{3}(\frac{1}{3}) + 12dx{\frac{1}{3}}^{x}{e}^{x}ln^{2}(\frac{1}{3}) + 4dx{\frac{1}{3}}^{x}{e}^{x}ln(\frac{1}{3}) + 12dx{e}^{x}{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 12dx{e}^{x}{\frac{1}{3}}^{x}ln(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x}ln^{3}(\frac{1}{3}) + 4dx{e}^{x}{\frac{1}{3}}^{x} + 216d{\frac{1}{3}}^{x}ln^{2}(\frac{1}{3}) + 144dx{\frac{1}{3}}^{x}ln^{3}(\frac{1}{3}) + 18dx^{2}{\frac{1}{3}}^{x}ln^{4}(\frac{1}{3})\\ \end{split}\end{equation} \]