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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ a(log_{b(log_{cx}^{d})}^{e^{log_{f}^{gx}}})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}\right)}{dx}\\=&a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})\\=&\frac{a}{xln(f)ln(blog_{cx}^{d})} + \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{xln(cx)ln(blog_{cx}^{d})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{a}{xln(f)ln(blog_{cx}^{d})} + \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{xln(cx)ln(blog_{cx}^{d})}\right)}{dx}\\=&\frac{a*-1}{x^{2}ln(f)ln(blog_{cx}^{d})} + \frac{a*-0}{xln^{2}(f)(f)ln(blog_{cx}^{d})} + \frac{a*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{xln(f)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{a*-log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln(cx)ln(blog_{cx}^{d})} + \frac{a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{xln(cx)ln(blog_{cx}^{d})} + \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-c}{xln^{2}(cx)(cx)ln(blog_{cx}^{d})} + \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{xln(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})}\\=&\frac{a}{x^{2}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} + \frac{a}{x^{2}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln(cx)ln(blog_{cx}^{d})} - \frac{a}{x^{2}ln(f)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln^{2}(cx)ln(blog_{cx}^{d})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{a}{x^{2}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} + \frac{a}{x^{2}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln(cx)ln(blog_{cx}^{d})} - \frac{a}{x^{2}ln(f)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln^{2}(cx)ln(blog_{cx}^{d})}\right)}{dx}\\=&\frac{a*-2}{x^{3}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} + \frac{a*-0}{x^{2}ln^{2}(f)(f)ln(cx)ln^{2}(blog_{cx}^{d})} + \frac{a*-c}{x^{2}ln(f)ln^{2}(cx)(cx)ln^{2}(blog_{cx}^{d})} + \frac{a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln(f)ln(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{a*-2}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} + \frac{a*-0}{x^{2}ln^{2}(f)(f)ln^{2}(blog_{cx}^{d})ln(cx)} + \frac{a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln(f)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})ln(cx)} + \frac{a*-c}{x^{2}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)(cx)} - \frac{a*-2log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln(cx)ln(blog_{cx}^{d})} - \frac{a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{2}ln(cx)ln(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-c}{x^{2}ln^{2}(cx)(cx)ln(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{a*-2}{x^{3}ln(f)ln(blog_{cx}^{d})} - \frac{a*-0}{x^{2}ln^{2}(f)(f)ln(blog_{cx}^{d})} - \frac{a*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln(f)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{2a*-2log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} + \frac{2a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{2}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2c}{x^{2}ln^{3}(cx)(cx)ln^{2}(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln^{2}(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{a*-2log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln(blog_{cx}^{d})} - \frac{a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{2}ln^{2}(cx)ln(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2c}{x^{2}ln^{3}(cx)(cx)ln(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln^{2}(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})}\\=& - \frac{3a}{x^{3}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} - \frac{a}{x^{3}ln^{2}(cx)ln(f)ln^{2}(blog_{cx}^{d})} + \frac{2a}{x^{3}ln^{2}(cx)ln(f)ln^{3}(blog_{cx}^{d})} - \frac{3a}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} + \frac{2a}{x^{3}ln(f)ln^{2}(cx)ln^{3}(blog_{cx}^{d})} - \frac{a}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})ln(f)} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln(cx)ln(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} + \frac{3alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln(blog_{cx}^{d})} + \frac{2a}{x^{3}ln(f)ln^{3}(blog_{cx}^{d})ln^{2}(cx)} - \frac{a}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)} + \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln^{3}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln^{2}(blog_{cx}^{d})} + \frac{2a}{x^{3}ln(f)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln(blog_{cx}^{d})}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{3a}{x^{3}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} - \frac{a}{x^{3}ln^{2}(cx)ln(f)ln^{2}(blog_{cx}^{d})} + \frac{2a}{x^{3}ln^{2}(cx)ln(f)ln^{3}(blog_{cx}^{d})} - \frac{3a}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} + \frac{2a}{x^{3}ln(f)ln^{2}(cx)ln^{3}(blog_{cx}^{d})} - \frac{a}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})ln(f)} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln(cx)ln(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} + \frac{3alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln(blog_{cx}^{d})} + \frac{2a}{x^{3}ln(f)ln^{3}(blog_{cx}^{d})ln^{2}(cx)} - \frac{a}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)} + \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln^{3}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln^{2}(blog_{cx}^{d})} + \frac{2a}{x^{3}ln(f)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln(blog_{cx}^{d})}\right)}{dx}\\=& - \frac{3a*-3}{x^{4}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} - \frac{3a*-0}{x^{3}ln^{2}(f)(f)ln(cx)ln^{2}(blog_{cx}^{d})} - \frac{3a*-c}{x^{3}ln(f)ln^{2}(cx)(cx)ln^{2}(blog_{cx}^{d})} - \frac{3a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{a*-3}{x^{4}ln^{2}(cx)ln(f)ln^{2}(blog_{cx}^{d})} - \frac{a*-2c}{x^{3}ln^{3}(cx)(cx)ln(f)ln^{2}(blog_{cx}^{d})} - \frac{a*-0}{x^{3}ln^{2}(cx)ln^{2}(f)(f)ln^{2}(blog_{cx}^{d})} - \frac{a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{2}(cx)ln(f)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{2a*-3}{x^{4}ln^{2}(cx)ln(f)ln^{3}(blog_{cx}^{d})} + \frac{2a*-2c}{x^{3}ln^{3}(cx)(cx)ln(f)ln^{3}(blog_{cx}^{d})} + \frac{2a*-0}{x^{3}ln^{2}(cx)ln^{2}(f)(f)ln^{3}(blog_{cx}^{d})} + \frac{2a*-3b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{2}(cx)ln(f)ln^{4}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{3a*-3}{x^{4}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} - \frac{3a*-0}{x^{3}ln^{2}(f)(f)ln^{2}(blog_{cx}^{d})ln(cx)} - \frac{3a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})ln(cx)} - \frac{3a*-c}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)(cx)} + \frac{2a*-3}{x^{4}ln(f)ln^{2}(cx)ln^{3}(blog_{cx}^{d})} + \frac{2a*-0}{x^{3}ln^{2}(f)(f)ln^{2}(cx)ln^{3}(blog_{cx}^{d})} + \frac{2a*-2c}{x^{3}ln(f)ln^{3}(cx)(cx)ln^{3}(blog_{cx}^{d})} + \frac{2a*-3b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln^{2}(cx)ln^{4}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{a*-3}{x^{4}ln^{2}(cx)ln^{2}(blog_{cx}^{d})ln(f)} - \frac{a*-2c}{x^{3}ln^{3}(cx)(cx)ln^{2}(blog_{cx}^{d})ln(f)} - \frac{a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{2}(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})ln(f)} - \frac{a*-0}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})ln^{2}(f)(f)} + \frac{2a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln(cx)ln(blog_{cx}^{d})} + \frac{2a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln(cx)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-c}{x^{3}ln^{2}(cx)(cx)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{6a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} - \frac{6a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2c}{x^{3}ln^{3}(cx)(cx)ln^{2}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{2}(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{3a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{2}(cx)ln(blog_{cx}^{d})} + \frac{3a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln^{2}(cx)ln(blog_{cx}^{d})} + \frac{3alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2c}{x^{3}ln^{3}(cx)(cx)ln(blog_{cx}^{d})} + \frac{3alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{2a*-3}{x^{4}ln(f)ln^{3}(blog_{cx}^{d})ln^{2}(cx)} + \frac{2a*-0}{x^{3}ln^{2}(f)(f)ln^{3}(blog_{cx}^{d})ln^{2}(cx)} + \frac{2a*-3b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln^{4}(blog_{cx}^{d})(blog_{cx}^{d})ln^{2}(cx)} + \frac{2a*-2c}{x^{3}ln(f)ln^{3}(blog_{cx}^{d})ln^{3}(cx)(cx)} - \frac{a*-3}{x^{4}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)} - \frac{a*-0}{x^{3}ln^{2}(f)(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)} - \frac{a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})ln^{2}(cx)} - \frac{a*-2c}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln^{3}(cx)(cx)} + \frac{6a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln^{3}(blog_{cx}^{d})} + \frac{6a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln^{3}(cx)ln^{3}(blog_{cx}^{d})} + \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-3c}{x^{3}ln^{4}(cx)(cx)ln^{3}(blog_{cx}^{d})} + \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-3b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{3}(cx)ln^{4}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{6a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln^{2}(blog_{cx}^{d})} - \frac{6a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln^{3}(cx)ln^{2}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-3c}{x^{3}ln^{4}(cx)(cx)ln^{2}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{3}(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{2a*-3}{x^{4}ln(f)ln(blog_{cx}^{d})} + \frac{2a*-0}{x^{3}ln^{2}(f)(f)ln(blog_{cx}^{d})} + \frac{2a*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{2a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln(blog_{cx}^{d})} + \frac{2a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln^{3}(cx)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-3c}{x^{3}ln^{4}(cx)(cx)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{3}(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})}\\=&\frac{11a}{x^{4}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} + \frac{6a}{x^{4}ln^{2}(cx)ln(f)ln^{2}(blog_{cx}^{d})} - \frac{12a}{x^{4}ln^{2}(cx)ln(f)ln^{3}(blog_{cx}^{d})} + \frac{2a}{x^{4}ln^{3}(cx)ln(f)ln^{2}(blog_{cx}^{d})} - \frac{4a}{x^{4}ln(f)ln^{3}(cx)ln^{3}(blog_{cx}^{d})} - \frac{8a}{x^{4}ln^{3}(cx)ln(f)ln^{3}(blog_{cx}^{d})} + \frac{12a}{x^{4}ln(f)ln^{3}(cx)ln^{4}(blog_{cx}^{d})} + \frac{11a}{x^{4}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} - \frac{12a}{x^{4}ln(f)ln^{2}(cx)ln^{3}(blog_{cx}^{d})} + \frac{6a}{x^{4}ln^{2}(cx)ln^{2}(blog_{cx}^{d})ln(f)} + \frac{6a}{x^{4}ln^{3}(cx)ln(f)ln^{4}(blog_{cx}^{d})} + \frac{4a}{x^{4}ln^{3}(cx)ln^{2}(blog_{cx}^{d})ln(f)} - \frac{6a}{x^{4}ln^{3}(cx)ln^{3}(blog_{cx}^{d})ln(f)} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln(cx)ln(blog_{cx}^{d})} + \frac{22alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} - \frac{11alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{2}(cx)ln(blog_{cx}^{d})} - \frac{12a}{x^{4}ln(f)ln^{3}(blog_{cx}^{d})ln^{2}(cx)} - \frac{36alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln^{3}(blog_{cx}^{d})} + \frac{36alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln^{2}(blog_{cx}^{d})} + \frac{6a}{x^{4}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)} - \frac{12alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln(blog_{cx}^{d})} + \frac{6a}{x^{4}ln(f)ln^{4}(blog_{cx}^{d})ln^{3}(cx)} + \frac{24alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{4}(cx)ln^{4}(blog_{cx}^{d})} - \frac{36alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{4}(cx)ln^{3}(blog_{cx}^{d})} - \frac{6a}{x^{4}ln(f)ln^{3}(blog_{cx}^{d})ln^{3}(cx)} + \frac{22alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{4}(cx)ln^{2}(blog_{cx}^{d})} + \frac{2a}{x^{4}ln(f)ln^{2}(blog_{cx}^{d})ln^{3}(cx)} - \frac{6a}{x^{4}ln(f)ln(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{4}(cx)ln(blog_{cx}^{d})}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ a(log_{b(log_{cx}^{d})}^{e^{log_{f}^{gx}}})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}\right)}{dx}\\=&a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})\\=&\frac{a}{xln(f)ln(blog_{cx}^{d})} + \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{xln(cx)ln(blog_{cx}^{d})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{a}{xln(f)ln(blog_{cx}^{d})} + \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{xln(cx)ln(blog_{cx}^{d})}\right)}{dx}\\=&\frac{a*-1}{x^{2}ln(f)ln(blog_{cx}^{d})} + \frac{a*-0}{xln^{2}(f)(f)ln(blog_{cx}^{d})} + \frac{a*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{xln(f)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{a*-log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln(cx)ln(blog_{cx}^{d})} + \frac{a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{xln(cx)ln(blog_{cx}^{d})} + \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-c}{xln^{2}(cx)(cx)ln(blog_{cx}^{d})} + \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{xln(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})}\\=&\frac{a}{x^{2}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} + \frac{a}{x^{2}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln(cx)ln(blog_{cx}^{d})} - \frac{a}{x^{2}ln(f)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln^{2}(cx)ln(blog_{cx}^{d})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{a}{x^{2}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} + \frac{a}{x^{2}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln(cx)ln(blog_{cx}^{d})} - \frac{a}{x^{2}ln(f)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{2}ln^{2}(cx)ln(blog_{cx}^{d})}\right)}{dx}\\=&\frac{a*-2}{x^{3}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} + \frac{a*-0}{x^{2}ln^{2}(f)(f)ln(cx)ln^{2}(blog_{cx}^{d})} + \frac{a*-c}{x^{2}ln(f)ln^{2}(cx)(cx)ln^{2}(blog_{cx}^{d})} + \frac{a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln(f)ln(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{a*-2}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} + \frac{a*-0}{x^{2}ln^{2}(f)(f)ln^{2}(blog_{cx}^{d})ln(cx)} + \frac{a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln(f)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})ln(cx)} + \frac{a*-c}{x^{2}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)(cx)} - \frac{a*-2log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln(cx)ln(blog_{cx}^{d})} - \frac{a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{2}ln(cx)ln(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-c}{x^{2}ln^{2}(cx)(cx)ln(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{a*-2}{x^{3}ln(f)ln(blog_{cx}^{d})} - \frac{a*-0}{x^{2}ln^{2}(f)(f)ln(blog_{cx}^{d})} - \frac{a*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln(f)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{2a*-2log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} + \frac{2a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{2}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2c}{x^{2}ln^{3}(cx)(cx)ln^{2}(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln^{2}(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{a*-2log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln(blog_{cx}^{d})} - \frac{a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{2}ln^{2}(cx)ln(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2c}{x^{2}ln^{3}(cx)(cx)ln(blog_{cx}^{d})} - \frac{alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{2}ln^{2}(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})}\\=& - \frac{3a}{x^{3}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} - \frac{a}{x^{3}ln^{2}(cx)ln(f)ln^{2}(blog_{cx}^{d})} + \frac{2a}{x^{3}ln^{2}(cx)ln(f)ln^{3}(blog_{cx}^{d})} - \frac{3a}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} + \frac{2a}{x^{3}ln(f)ln^{2}(cx)ln^{3}(blog_{cx}^{d})} - \frac{a}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})ln(f)} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln(cx)ln(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} + \frac{3alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln(blog_{cx}^{d})} + \frac{2a}{x^{3}ln(f)ln^{3}(blog_{cx}^{d})ln^{2}(cx)} - \frac{a}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)} + \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln^{3}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln^{2}(blog_{cx}^{d})} + \frac{2a}{x^{3}ln(f)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln(blog_{cx}^{d})}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{3a}{x^{3}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} - \frac{a}{x^{3}ln^{2}(cx)ln(f)ln^{2}(blog_{cx}^{d})} + \frac{2a}{x^{3}ln^{2}(cx)ln(f)ln^{3}(blog_{cx}^{d})} - \frac{3a}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} + \frac{2a}{x^{3}ln(f)ln^{2}(cx)ln^{3}(blog_{cx}^{d})} - \frac{a}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})ln(f)} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln(cx)ln(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} + \frac{3alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{2}(cx)ln(blog_{cx}^{d})} + \frac{2a}{x^{3}ln(f)ln^{3}(blog_{cx}^{d})ln^{2}(cx)} - \frac{a}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)} + \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln^{3}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln^{2}(blog_{cx}^{d})} + \frac{2a}{x^{3}ln(f)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{3}ln^{3}(cx)ln(blog_{cx}^{d})}\right)}{dx}\\=& - \frac{3a*-3}{x^{4}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} - \frac{3a*-0}{x^{3}ln^{2}(f)(f)ln(cx)ln^{2}(blog_{cx}^{d})} - \frac{3a*-c}{x^{3}ln(f)ln^{2}(cx)(cx)ln^{2}(blog_{cx}^{d})} - \frac{3a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{a*-3}{x^{4}ln^{2}(cx)ln(f)ln^{2}(blog_{cx}^{d})} - \frac{a*-2c}{x^{3}ln^{3}(cx)(cx)ln(f)ln^{2}(blog_{cx}^{d})} - \frac{a*-0}{x^{3}ln^{2}(cx)ln^{2}(f)(f)ln^{2}(blog_{cx}^{d})} - \frac{a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{2}(cx)ln(f)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{2a*-3}{x^{4}ln^{2}(cx)ln(f)ln^{3}(blog_{cx}^{d})} + \frac{2a*-2c}{x^{3}ln^{3}(cx)(cx)ln(f)ln^{3}(blog_{cx}^{d})} + \frac{2a*-0}{x^{3}ln^{2}(cx)ln^{2}(f)(f)ln^{3}(blog_{cx}^{d})} + \frac{2a*-3b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{2}(cx)ln(f)ln^{4}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{3a*-3}{x^{4}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} - \frac{3a*-0}{x^{3}ln^{2}(f)(f)ln^{2}(blog_{cx}^{d})ln(cx)} - \frac{3a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})ln(cx)} - \frac{3a*-c}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)(cx)} + \frac{2a*-3}{x^{4}ln(f)ln^{2}(cx)ln^{3}(blog_{cx}^{d})} + \frac{2a*-0}{x^{3}ln^{2}(f)(f)ln^{2}(cx)ln^{3}(blog_{cx}^{d})} + \frac{2a*-2c}{x^{3}ln(f)ln^{3}(cx)(cx)ln^{3}(blog_{cx}^{d})} + \frac{2a*-3b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln^{2}(cx)ln^{4}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{a*-3}{x^{4}ln^{2}(cx)ln^{2}(blog_{cx}^{d})ln(f)} - \frac{a*-2c}{x^{3}ln^{3}(cx)(cx)ln^{2}(blog_{cx}^{d})ln(f)} - \frac{a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{2}(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})ln(f)} - \frac{a*-0}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})ln^{2}(f)(f)} + \frac{2a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln(cx)ln(blog_{cx}^{d})} + \frac{2a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln(cx)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-c}{x^{3}ln^{2}(cx)(cx)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{6a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} - \frac{6a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2c}{x^{3}ln^{3}(cx)(cx)ln^{2}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{2}(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{3a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{2}(cx)ln(blog_{cx}^{d})} + \frac{3a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln^{2}(cx)ln(blog_{cx}^{d})} + \frac{3alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2c}{x^{3}ln^{3}(cx)(cx)ln(blog_{cx}^{d})} + \frac{3alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{2}(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{2a*-3}{x^{4}ln(f)ln^{3}(blog_{cx}^{d})ln^{2}(cx)} + \frac{2a*-0}{x^{3}ln^{2}(f)(f)ln^{3}(blog_{cx}^{d})ln^{2}(cx)} + \frac{2a*-3b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln^{4}(blog_{cx}^{d})(blog_{cx}^{d})ln^{2}(cx)} + \frac{2a*-2c}{x^{3}ln(f)ln^{3}(blog_{cx}^{d})ln^{3}(cx)(cx)} - \frac{a*-3}{x^{4}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)} - \frac{a*-0}{x^{3}ln^{2}(f)(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)} - \frac{a*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})ln^{2}(cx)} - \frac{a*-2c}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})ln^{3}(cx)(cx)} + \frac{6a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln^{3}(blog_{cx}^{d})} + \frac{6a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln^{3}(cx)ln^{3}(blog_{cx}^{d})} + \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-3c}{x^{3}ln^{4}(cx)(cx)ln^{3}(blog_{cx}^{d})} + \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-3b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{3}(cx)ln^{4}(blog_{cx}^{d})(blog_{cx}^{d})} - \frac{6a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln^{2}(blog_{cx}^{d})} - \frac{6a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln^{3}(cx)ln^{2}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-3c}{x^{3}ln^{4}(cx)(cx)ln^{2}(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-2b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{3}(cx)ln^{3}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{2a*-3}{x^{4}ln(f)ln(blog_{cx}^{d})} + \frac{2a*-0}{x^{3}ln^{2}(f)(f)ln(blog_{cx}^{d})} + \frac{2a*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln(f)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})} + \frac{2a*-3log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln(blog_{cx}^{d})} + \frac{2a(\frac{(\frac{(e^{log_{f}^{gx}}(\frac{(\frac{(g)}{(gx)} - \frac{(0)log_{f}^{gx}}{(f)})}{(ln(f))}))}{(e^{log_{f}^{gx}})} - \frac{(b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))}))log_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{(blog_{cx}^{d})})}{(ln(blog_{cx}^{d}))})}{x^{3}ln^{3}(cx)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-3c}{x^{3}ln^{4}(cx)(cx)ln(blog_{cx}^{d})} + \frac{2alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}*-b(\frac{(\frac{(0)}{(d)} - \frac{(c)log_{cx}^{d}}{(cx)})}{(ln(cx))})}{x^{3}ln^{3}(cx)ln^{2}(blog_{cx}^{d})(blog_{cx}^{d})}\\=&\frac{11a}{x^{4}ln(f)ln(cx)ln^{2}(blog_{cx}^{d})} + \frac{6a}{x^{4}ln^{2}(cx)ln(f)ln^{2}(blog_{cx}^{d})} - \frac{12a}{x^{4}ln^{2}(cx)ln(f)ln^{3}(blog_{cx}^{d})} + \frac{2a}{x^{4}ln^{3}(cx)ln(f)ln^{2}(blog_{cx}^{d})} - \frac{4a}{x^{4}ln(f)ln^{3}(cx)ln^{3}(blog_{cx}^{d})} - \frac{8a}{x^{4}ln^{3}(cx)ln(f)ln^{3}(blog_{cx}^{d})} + \frac{12a}{x^{4}ln(f)ln^{3}(cx)ln^{4}(blog_{cx}^{d})} + \frac{11a}{x^{4}ln(f)ln^{2}(blog_{cx}^{d})ln(cx)} - \frac{12a}{x^{4}ln(f)ln^{2}(cx)ln^{3}(blog_{cx}^{d})} + \frac{6a}{x^{4}ln^{2}(cx)ln^{2}(blog_{cx}^{d})ln(f)} + \frac{6a}{x^{4}ln^{3}(cx)ln(f)ln^{4}(blog_{cx}^{d})} + \frac{4a}{x^{4}ln^{3}(cx)ln^{2}(blog_{cx}^{d})ln(f)} - \frac{6a}{x^{4}ln^{3}(cx)ln^{3}(blog_{cx}^{d})ln(f)} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln(cx)ln(blog_{cx}^{d})} + \frac{22alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{2}(cx)ln^{2}(blog_{cx}^{d})} - \frac{11alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{2}(cx)ln(blog_{cx}^{d})} - \frac{12a}{x^{4}ln(f)ln^{3}(blog_{cx}^{d})ln^{2}(cx)} - \frac{36alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln^{3}(blog_{cx}^{d})} + \frac{36alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln^{2}(blog_{cx}^{d})} + \frac{6a}{x^{4}ln(f)ln^{2}(blog_{cx}^{d})ln^{2}(cx)} - \frac{12alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{3}(cx)ln(blog_{cx}^{d})} + \frac{6a}{x^{4}ln(f)ln^{4}(blog_{cx}^{d})ln^{3}(cx)} + \frac{24alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{4}(cx)ln^{4}(blog_{cx}^{d})} - \frac{36alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{4}(cx)ln^{3}(blog_{cx}^{d})} - \frac{6a}{x^{4}ln(f)ln^{3}(blog_{cx}^{d})ln^{3}(cx)} + \frac{22alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{4}(cx)ln^{2}(blog_{cx}^{d})} + \frac{2a}{x^{4}ln(f)ln^{2}(blog_{cx}^{d})ln^{3}(cx)} - \frac{6a}{x^{4}ln(f)ln(blog_{cx}^{d})} - \frac{6alog_{blog_{cx}^{d}}^{e^{log_{f}^{gx}}}}{x^{4}ln^{4}(cx)ln(blog_{cx}^{d})}\\ \end{split}\end{equation} \]