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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ (xx + x + 11)sqrt(xxx + 5x + 121)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = x^{2}sqrt(x^{3} + 5x + 121) + xsqrt(x^{3} + 5x + 121) + 11sqrt(x^{3} + 5x + 121)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( x^{2}sqrt(x^{3} + 5x + 121) + xsqrt(x^{3} + 5x + 121) + 11sqrt(x^{3} + 5x + 121)\right)}{dx}\\=&2xsqrt(x^{3} + 5x + 121) + \frac{x^{2}(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + sqrt(x^{3} + 5x + 121) + \frac{x(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{11(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}}\\=&2xsqrt(x^{3} + 5x + 121) + \frac{3x^{4}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{19x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + sqrt(x^{3} + 5x + 121) + \frac{3x^{3}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{5x}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{55}{2(x^{3} + 5x + 121)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2xsqrt(x^{3} + 5x + 121) + \frac{3x^{4}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{19x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + sqrt(x^{3} + 5x + 121) + \frac{3x^{3}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{5x}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{55}{2(x^{3} + 5x + 121)^{\frac{1}{2}}}\right)}{dx}\\=&2sqrt(x^{3} + 5x + 121) + \frac{2x(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{3(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{4}}{2} + \frac{3*4x^{3}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + 19(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{2} + \frac{19*2x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{3(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{3}}{2} + \frac{3*3x^{2}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{5(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x}{2} + \frac{5}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{55(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})}{2}\\=&2sqrt(x^{3} + 5x + 121) + \frac{9x^{3}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{43x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{9x^{6}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{129x^{4}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{9x^{5}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{15x^{3}}{2(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{6x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{355x^{2}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{25x}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{275}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{5}{(x^{3} + 5x + 121)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2sqrt(x^{3} + 5x + 121) + \frac{9x^{3}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{43x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{9x^{6}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{129x^{4}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{9x^{5}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{15x^{3}}{2(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{6x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{355x^{2}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{25x}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{275}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{5}{(x^{3} + 5x + 121)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{2(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + 9(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{3} + \frac{9*3x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + 43(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x + \frac{43}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{9(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{6}}{4} - \frac{9*6x^{5}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{129(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{4}}{4} - \frac{129*4x^{3}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{9(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{5}}{4} - \frac{9*5x^{4}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{15(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{3}}{2} - \frac{15*3x^{2}}{2(x^{3} + 5x + 121)^{\frac{3}{2}}} + 6(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{2} + \frac{6*2x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{355(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{2}}{4} - \frac{355*2x}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{25(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x}{4} - \frac{25}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{275(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})}{4} + 5(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})\\=&\frac{30x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{81x^{8}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{27x^{5}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{216x^{3}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{162x^{6}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{81x^{7}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{405x^{5}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{2565x^{4}}{4(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{81x^{4}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{675x^{3}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{45x^{2}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{285x}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{12x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{975x^{2}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{375x}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{48}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{4125}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{75}{4(x^{3} + 5x + 121)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{30x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{81x^{8}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{27x^{5}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{216x^{3}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{162x^{6}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{81x^{7}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{405x^{5}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{2565x^{4}}{4(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{81x^{4}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{675x^{3}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{45x^{2}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{285x}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{12x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{975x^{2}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{375x}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{48}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{4125}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{75}{4(x^{3} + 5x + 121)^{\frac{3}{2}}}\right)}{dx}\\=&30(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{2} + \frac{30*2x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{81(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{8}}{8} + \frac{81*8x^{7}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - 27(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{5} - \frac{27*5x^{4}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - 216(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{3} - \frac{216*3x^{2}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + 162(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{6} + \frac{162*6x^{5}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{81(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{7}}{8} + \frac{81*7x^{6}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{405(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{5}}{8} + \frac{405*5x^{4}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{2565(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{4}}{4} + \frac{2565*4x^{3}}{4(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{81(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{4}}{4} - \frac{81*4x^{3}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{675(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{3}}{8} + \frac{675*3x^{2}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - 45(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{2} - \frac{45*2x}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - 285(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x - \frac{285}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + 12(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x + \frac{12}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + 975(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{2} + \frac{975*2x}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{375(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x}{8} + \frac{375}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + 48(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}}) + \frac{4125(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})}{8} - \frac{75(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})}{4}\\=&\frac{-180x^{4}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{1215x^{10}}{16(x^{3} + 5x + 121)^{\frac{7}{2}}} + \frac{60x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{405x^{7}}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{4293x^{5}}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{21465x^{8}}{16(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{1215x^{9}}{16(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{2025x^{7}}{4(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{54675x^{6}}{8(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{795x^{2}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{162x^{6}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{10125x^{5}}{8(x^{3} + 5x + 121)^{\frac{7}{2}}} + \frac{1215x^{4}}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{10935x^{3}}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{99x^{3}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{122625x^{4}}{8(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{5625x^{3}}{4(x^{3} + 5x + 121)^{\frac{7}{2}}} + \frac{675x^{2}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{120x}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{8175x}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{256875x^{2}}{16(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{9375x}{16(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{405}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{12}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{375}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{103125}{16(x^{3} + 5x + 121)^{\frac{7}{2}}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ (xx + x + 11)sqrt(xxx + 5x + 121)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = x^{2}sqrt(x^{3} + 5x + 121) + xsqrt(x^{3} + 5x + 121) + 11sqrt(x^{3} + 5x + 121)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( x^{2}sqrt(x^{3} + 5x + 121) + xsqrt(x^{3} + 5x + 121) + 11sqrt(x^{3} + 5x + 121)\right)}{dx}\\=&2xsqrt(x^{3} + 5x + 121) + \frac{x^{2}(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + sqrt(x^{3} + 5x + 121) + \frac{x(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{11(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}}\\=&2xsqrt(x^{3} + 5x + 121) + \frac{3x^{4}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{19x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + sqrt(x^{3} + 5x + 121) + \frac{3x^{3}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{5x}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{55}{2(x^{3} + 5x + 121)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2xsqrt(x^{3} + 5x + 121) + \frac{3x^{4}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{19x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + sqrt(x^{3} + 5x + 121) + \frac{3x^{3}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{5x}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{55}{2(x^{3} + 5x + 121)^{\frac{1}{2}}}\right)}{dx}\\=&2sqrt(x^{3} + 5x + 121) + \frac{2x(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{3(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{4}}{2} + \frac{3*4x^{3}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + 19(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{2} + \frac{19*2x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{3(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{3}}{2} + \frac{3*3x^{2}}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{5(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x}{2} + \frac{5}{2(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{55(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})}{2}\\=&2sqrt(x^{3} + 5x + 121) + \frac{9x^{3}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{43x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{9x^{6}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{129x^{4}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{9x^{5}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{15x^{3}}{2(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{6x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{355x^{2}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{25x}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{275}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{5}{(x^{3} + 5x + 121)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2sqrt(x^{3} + 5x + 121) + \frac{9x^{3}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{43x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{9x^{6}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{129x^{4}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{9x^{5}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{15x^{3}}{2(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{6x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{355x^{2}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{25x}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{275}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{5}{(x^{3} + 5x + 121)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{2(3x^{2} + 5 + 0)*\frac{1}{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + 9(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{3} + \frac{9*3x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + 43(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x + \frac{43}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{9(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{6}}{4} - \frac{9*6x^{5}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{129(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{4}}{4} - \frac{129*4x^{3}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{9(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{5}}{4} - \frac{9*5x^{4}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{15(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{3}}{2} - \frac{15*3x^{2}}{2(x^{3} + 5x + 121)^{\frac{3}{2}}} + 6(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{2} + \frac{6*2x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} - \frac{355(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{2}}{4} - \frac{355*2x}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{25(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x}{4} - \frac{25}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{275(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})}{4} + 5(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})\\=&\frac{30x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{81x^{8}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{27x^{5}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{216x^{3}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{162x^{6}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{81x^{7}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{405x^{5}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{2565x^{4}}{4(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{81x^{4}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{675x^{3}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{45x^{2}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{285x}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{12x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{975x^{2}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{375x}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{48}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{4125}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{75}{4(x^{3} + 5x + 121)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{30x^{2}}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{81x^{8}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{27x^{5}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{216x^{3}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{162x^{6}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{81x^{7}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{405x^{5}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{2565x^{4}}{4(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{81x^{4}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{675x^{3}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{45x^{2}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{285x}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{12x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{975x^{2}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{375x}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{48}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{4125}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{75}{4(x^{3} + 5x + 121)^{\frac{3}{2}}}\right)}{dx}\\=&30(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x^{2} + \frac{30*2x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{81(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{8}}{8} + \frac{81*8x^{7}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - 27(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{5} - \frac{27*5x^{4}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - 216(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{3} - \frac{216*3x^{2}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + 162(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{6} + \frac{162*6x^{5}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{81(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{7}}{8} + \frac{81*7x^{6}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{405(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{5}}{8} + \frac{405*5x^{4}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{2565(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{4}}{4} + \frac{2565*4x^{3}}{4(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{81(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{4}}{4} - \frac{81*4x^{3}}{4(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{675(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{3}}{8} + \frac{675*3x^{2}}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} - 45(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x^{2} - \frac{45*2x}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - 285(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})x - \frac{285}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + 12(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}})x + \frac{12}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + 975(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x^{2} + \frac{975*2x}{(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{375(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})x}{8} + \frac{375}{8(x^{3} + 5x + 121)^{\frac{5}{2}}} + 48(\frac{\frac{-1}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{3}{2}}}) + \frac{4125(\frac{\frac{-5}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{7}{2}}})}{8} - \frac{75(\frac{\frac{-3}{2}(3x^{2} + 5 + 0)}{(x^{3} + 5x + 121)^{\frac{5}{2}}})}{4}\\=&\frac{-180x^{4}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{1215x^{10}}{16(x^{3} + 5x + 121)^{\frac{7}{2}}} + \frac{60x}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{405x^{7}}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{4293x^{5}}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{21465x^{8}}{16(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{1215x^{9}}{16(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{2025x^{7}}{4(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{54675x^{6}}{8(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{795x^{2}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{162x^{6}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{10125x^{5}}{8(x^{3} + 5x + 121)^{\frac{7}{2}}} + \frac{1215x^{4}}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} + \frac{10935x^{3}}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{99x^{3}}{(x^{3} + 5x + 121)^{\frac{3}{2}}} - \frac{122625x^{4}}{8(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{5625x^{3}}{4(x^{3} + 5x + 121)^{\frac{7}{2}}} + \frac{675x^{2}}{(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{120x}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{8175x}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{256875x^{2}}{16(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{9375x}{16(x^{3} + 5x + 121)^{\frac{7}{2}}} - \frac{405}{(x^{3} + 5x + 121)^{\frac{3}{2}}} + \frac{12}{(x^{3} + 5x + 121)^{\frac{1}{2}}} + \frac{375}{2(x^{3} + 5x + 121)^{\frac{5}{2}}} - \frac{103125}{16(x^{3} + 5x + 121)^{\frac{7}{2}}}\\ \end{split}\end{equation} \]