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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {({x}^{2} + 1)}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (x^{2} + 1)^{x}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (x^{2} + 1)^{x}\right)}{dx}\\=&((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))\\=&(x^{2} + 1)^{x}ln(x^{2} + 1) + \frac{2x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( (x^{2} + 1)^{x}ln(x^{2} + 1) + \frac{2x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)}\right)}{dx}\\=&((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln(x^{2} + 1) + \frac{(x^{2} + 1)^{x}(2x + 0)}{(x^{2} + 1)} + 2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}(x^{2} + 1)^{x} + \frac{2*2x(x^{2} + 1)^{x}}{(x^{2} + 1)} + \frac{2x^{2}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)}\\=&(x^{2} + 1)^{x}ln^{2}(x^{2} + 1) + \frac{4x^{2}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{6x(x^{2} + 1)^{x}}{(x^{2} + 1)} - \frac{4x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{4x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( (x^{2} + 1)^{x}ln^{2}(x^{2} + 1) + \frac{4x^{2}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{6x(x^{2} + 1)^{x}}{(x^{2} + 1)} - \frac{4x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{4x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}}\right)}{dx}\\=&((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln^{2}(x^{2} + 1) + \frac{(x^{2} + 1)^{x}*2ln(x^{2} + 1)(2x + 0)}{(x^{2} + 1)} + 4(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}(x^{2} + 1)^{x}ln(x^{2} + 1) + \frac{4*2x(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{4x^{2}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{4x^{2}(x^{2} + 1)^{x}(2x + 0)}{(x^{2} + 1)(x^{2} + 1)} + 6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x(x^{2} + 1)^{x} + \frac{6(x^{2} + 1)^{x}}{(x^{2} + 1)} + \frac{6x((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)} - 4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}(x^{2} + 1)^{x} - \frac{4*3x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} - \frac{4x^{3}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{2}} + 4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{4}(x^{2} + 1)^{x} + \frac{4*4x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{4x^{4}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{2}}\\=&(x^{2} + 1)^{x}ln^{3}(x^{2} + 1) + \frac{6x^{2}(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)} + \frac{18x(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} - \frac{12x^{3}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{12x^{4}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{36x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} - \frac{24x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{6(x^{2} + 1)^{x}}{(x^{2} + 1)} + \frac{16x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} - \frac{24x^{5}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{8x^{6}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( (x^{2} + 1)^{x}ln^{3}(x^{2} + 1) + \frac{6x^{2}(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)} + \frac{18x(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} - \frac{12x^{3}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{12x^{4}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{36x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} - \frac{24x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{6(x^{2} + 1)^{x}}{(x^{2} + 1)} + \frac{16x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} - \frac{24x^{5}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{8x^{6}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}}\right)}{dx}\\=&((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln^{3}(x^{2} + 1) + \frac{(x^{2} + 1)^{x}*3ln^{2}(x^{2} + 1)(2x + 0)}{(x^{2} + 1)} + 6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}(x^{2} + 1)^{x}ln^{2}(x^{2} + 1) + \frac{6*2x(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)} + \frac{6x^{2}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln^{2}(x^{2} + 1)}{(x^{2} + 1)} + \frac{6x^{2}(x^{2} + 1)^{x}*2ln(x^{2} + 1)(2x + 0)}{(x^{2} + 1)(x^{2} + 1)} + 18(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x(x^{2} + 1)^{x}ln(x^{2} + 1) + \frac{18(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{18x((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{18x(x^{2} + 1)^{x}(2x + 0)}{(x^{2} + 1)(x^{2} + 1)} - 12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}(x^{2} + 1)^{x}ln(x^{2} + 1) - \frac{12*3x^{2}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} - \frac{12x^{3}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln(x^{2} + 1)}{(x^{2} + 1)^{2}} - \frac{12x^{3}(x^{2} + 1)^{x}(2x + 0)}{(x^{2} + 1)^{2}(x^{2} + 1)} + 12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{4}(x^{2} + 1)^{x}ln(x^{2} + 1) + \frac{12*4x^{3}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{12x^{4}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{12x^{4}(x^{2} + 1)^{x}(2x + 0)}{(x^{2} + 1)^{2}(x^{2} + 1)} + 36(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}(x^{2} + 1)^{x} + \frac{36*3x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{36x^{3}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{2}} - 24(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}(x^{2} + 1)^{x} - \frac{24*2x(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} - \frac{24x^{2}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{2}} + 6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})(x^{2} + 1)^{x} + \frac{6((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)} + 16(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{4}(x^{2} + 1)^{x} + \frac{16*4x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{16x^{4}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{3}} - 24(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{5}(x^{2} + 1)^{x} - \frac{24*5x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} - \frac{24x^{5}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{3}} + 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{6}(x^{2} + 1)^{x} + \frac{8*6x^{5}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{8x^{6}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{3}}\\=&(x^{2} + 1)^{x}ln^{4}(x^{2} + 1) + \frac{8x^{2}(x^{2} + 1)^{x}ln^{3}(x^{2} + 1)}{(x^{2} + 1)} + \frac{36x(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)} - \frac{24x^{3}(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{24x^{4}(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{144x^{3}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} - \frac{96x^{2}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{24(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{64x^{4}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{3}} - \frac{96x^{5}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{3}} + \frac{32x^{6}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{3}} - \frac{336x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{156x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{144x^{5}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{160x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} - \frac{60x(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} - \frac{96x^{5}(x^{2} + 1)^{x}}{(x^{2} + 1)^{4}} + \frac{176x^{6}(x^{2} + 1)^{x}}{(x^{2} + 1)^{4}} - \frac{96x^{7}(x^{2} + 1)^{x}}{(x^{2} + 1)^{4}} + \frac{16x^{8}(x^{2} + 1)^{x}}{(x^{2} + 1)^{4}}\\ \end{split}\end{equation} \]