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 2 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ {(\frac{({2}^{x} + {53}^{x} + {49069}^{x})}{3})}^{(\frac{3}{x})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}\right)}{dx}\\=&((\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}((\frac{3*-1}{x^{2}})ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x}) + \frac{(\frac{3}{x})(\frac{1}{3}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})) + \frac{1}{3}({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)})) + \frac{1}{3}({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)})))}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}))\\=&\frac{-3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}{x^{2}} + \frac{{2}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{53}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{49069}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}{x^{2}} + \frac{{2}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{53}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{49069}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x}\right)}{dx}\\=&\frac{-3*-2(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}{x^{3}} - \frac{3((\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}((\frac{3*-1}{x^{2}})ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x}) + \frac{(\frac{3}{x})(\frac{1}{3}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})) + \frac{1}{3}({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)})) + \frac{1}{3}({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)})))}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}))ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}{x^{2}} - \frac{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}(\frac{1}{3}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})) + \frac{1}{3}({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)})) + \frac{1}{3}({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)})))}{x^{2}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})} + \frac{(\frac{-(\frac{1}{3}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})) + \frac{1}{3}({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)})) + \frac{1}{3}({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)})))}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}}){2}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)}{x} + \frac{-{2}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{2}} + \frac{({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{2}^{x}((\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}((\frac{3*-1}{x^{2}})ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x}) + \frac{(\frac{3}{x})(\frac{1}{3}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})) + \frac{1}{3}({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)})) + \frac{1}{3}({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)})))}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}))ln(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{2}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}*0}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x(2)} + \frac{(\frac{-(\frac{1}{3}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})) + \frac{1}{3}({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)})) + \frac{1}{3}({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)})))}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}}){53}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)}{x} + \frac{-{53}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{2}} + \frac{({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)}))(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{53}^{x}((\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}((\frac{3*-1}{x^{2}})ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x}) + \frac{(\frac{3}{x})(\frac{1}{3}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})) + \frac{1}{3}({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)})) + \frac{1}{3}({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)})))}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}))ln(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{53}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}*0}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x(53)} + \frac{(\frac{-(\frac{1}{3}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})) + \frac{1}{3}({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)})) + \frac{1}{3}({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)})))}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}}){49069}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)}{x} + \frac{-{49069}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{2}} + \frac{({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)}))(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{49069}^{x}((\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}((\frac{3*-1}{x^{2}})ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x}) + \frac{(\frac{3}{x})(\frac{1}{3}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})) + \frac{1}{3}({53}^{x}((1)ln(53) + \frac{(x)(0)}{(53)})) + \frac{1}{3}({49069}^{x}((1)ln(49069) + \frac{(x)(0)}{(49069)})))}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}))ln(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} + \frac{{49069}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}*0}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x(49069)}\\=&\frac{6(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}{x^{3}} + \frac{9(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln^{2}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}{x^{4}} - \frac{3 * {2}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{3}} - \frac{3 * {53}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{3}} - \frac{3 * {49069}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{3}} - \frac{{53}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)ln(2)}{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x} - \frac{{49069}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)ln(2)}{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x} - \frac{{2}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)ln(53)}{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x} - \frac{{49069}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)ln(53)}{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x} - \frac{{2}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)ln(49069)}{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x} - \frac{{53}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)ln(49069)}{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x} - \frac{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}{2}^{x}ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})ln(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{3}} + \frac{{53}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)ln(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x^{2}} + \frac{{49069}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)ln(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x^{2}} - \frac{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}{53}^{x}ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})ln(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{3}} + \frac{{2}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)ln(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x^{2}} + \frac{{49069}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)ln(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x^{2}} - \frac{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}{49069}^{x}ln(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})ln(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{3}} + \frac{{2}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)ln(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x^{2}} + \frac{{53}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)ln(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x^{2}} - \frac{{49069}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln^{2}(49069)}{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x} + \frac{{49069}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln^{2}(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} - \frac{2 * {53}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{2}} + \frac{{2}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln^{2}(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} - \frac{2 * {2}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{2}} + \frac{{53}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln^{2}(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x^{2}} + \frac{{53}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln^{2}(53)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x} - \frac{2 * {49069}^{x}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})x^{2}} - \frac{{53}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln^{2}(53)}{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x} + \frac{{2}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln^{2}(2)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x^{2}} - \frac{{2}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln^{2}(2)}{3(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x} + \frac{{49069}^{(2x)}(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{(\frac{3}{x})}ln^{2}(49069)}{(\frac{1}{3} * {2}^{x} + \frac{1}{3} * {53}^{x} + \frac{1}{3} * {49069}^{x})^{2}x^{2}}\\ \end{split}\end{equation} \]