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 5 questions in this calculation: for each question, the 4 derivative of ^ is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/5]Find\ the\ 4th\ derivative\ of\ function\ x + ^ + x(ln(x + ^ + 2))\ with\ respect\ to\ ^：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = xln(x + ^ + 2) + ^ + x\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xln(x + ^ + 2) + ^ + x\right)}{d^}\\=&\frac{x(0 + 1 + 0)}{(x + ^ + 2)} + 1 + 0\\=&\frac{x}{(x + ^ + 2)} + 1\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{x}{(x + ^ + 2)} + 1\right)}{d^}\\=&(\frac{-(0 + 1 + 0)}{(x + ^ + 2)^{2}})x + 0 + 0\\=&\frac{-x}{(x + ^ + 2)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-x}{(x + ^ + 2)^{2}}\right)}{d^}\\=&-(\frac{-2(0 + 1 + 0)}{(x + ^ + 2)^{3}})x + 0\\=&\frac{2x}{(x + ^ + 2)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2x}{(x + ^ + 2)^{3}}\right)}{d^}\\=&2(\frac{-3(0 + 1 + 0)}{(x + ^ + 2)^{4}})x + 0\\=&\frac{-6x}{(x + ^ + 2)^{4}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/5]Find\ the\ 4th\ derivative\ of\ function\ sin(e^{x}cos(x))\ with\ respect\ to\ ^：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(e^{x}cos(x))\right)}{d^}\\=&cos(e^{x}cos(x))(e^{x}*0cos(x) + e^{x}*-sin(x)*0)\\=&0\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d^}\\=&0\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d^}\\=&0\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d^}\\=&0\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/5]Find\ the\ 4th\ derivative\ of\ function\ cos(2 + ^ + (x(^(2)) + ^ + 3))\ with\ respect\ to\ ^：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = cos(2^ + x(^(2)) + 5)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(2^ + x(^(2)) + 5)\right)}{d^}\\=&-sin(2^ + x(^(2)) + 5)(2 + 0 + 0)\\=&-2sin(2^ + x(^(2)) + 5)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -2sin(2^ + x(^(2)) + 5)\right)}{d^}\\=&-2cos(2^ + x(^(2)) + 5)(2 + 0 + 0)\\=&-4cos(2^ + x(^(2)) + 5)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -4cos(2^ + x(^(2)) + 5)\right)}{d^}\\=&-4*-sin(2^ + x(^(2)) + 5)(2 + 0 + 0)\\=&8sin(2^ + x(^(2)) + 5)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 8sin(2^ + x(^(2)) + 5)\right)}{d^}\\=&8cos(2^ + x(^(2)) + 5)(2 + 0 + 0)\\=&16cos(2^ + x(^(2)) + 5)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/5]Find\ the\ 4th\ derivative\ of\ function\ arctan(x + ^ + 2 + x + ^ + 1 + 1)\ with\ respect\ to\ ^：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = arctan(2x + 2^ + 4)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arctan(2x + 2^ + 4)\right)}{d^}\\=&(\frac{(0 + 2 + 0)}{(1 + (2x + 2^ + 4)^{2})})\\=&\frac{2}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)}\right)}{d^}\\=&2(\frac{-(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{2}})\\=&\frac{-16x}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{2}} - \frac{16^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{2}} - \frac{32}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-16x}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{2}} - \frac{16^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{2}} - \frac{32}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{2}}\right)}{d^}\\=&-16(\frac{-2(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}})x + 0 - 16(\frac{-2(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}})^ - \frac{16}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{2}} - 32(\frac{-2(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}})\\=&\frac{512x^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} + \frac{256x^{2}}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} + \frac{1024x}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} + \frac{256^^{2}}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} + \frac{1024^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} - \frac{16}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{2}} + \frac{1024}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{512x^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} + \frac{256x^{2}}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} + \frac{1024x}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} + \frac{256^^{2}}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} + \frac{1024^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} - \frac{16}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{2}} + \frac{1024}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}}\right)}{d^}\\=&512(\frac{-3(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}})x^ + \frac{512x}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} + 256(\frac{-3(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}})x^{2} + 0 + 1024(\frac{-3(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}})x + 0 + 256(\frac{-3(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}})^^{2} + \frac{256*2^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} + 1024(\frac{-3(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}})^ + \frac{1024}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} - 16(\frac{-2(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}}) + 1024(\frac{-3(8x + 0 + 0 + 4*2^ + 16 + 0)}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}})\\=&\frac{-18432x^{2}^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}} - \frac{18432x^^{2}}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}} - \frac{73728x^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}} + \frac{768x}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} - \frac{6144x^{3}}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}} - \frac{36864x^{2}}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}} - \frac{6144^^{3}}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}} - \frac{36864^^{2}}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}} + \frac{768^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} - \frac{73728x}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}} - \frac{73728^}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}} + \frac{1536}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{3}} - \frac{49152}{(8x^ + 4x^{2} + 16x + 4^^{2} + 16^ + 17)^{4}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[5/5]Find\ the\ 4th\ derivative\ of\ function\ ln(1 + ^ + (2(^((3(^((4(^(x + 1) + 1) + 1) + 1)))))))\ with\ respect\ to\ ^：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(^ + 3)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(^ + 3)\right)}{d^}\\=&\frac{(1 + 0)}{(^ + 3)}\\=&\frac{1}{(^ + 3)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{(^ + 3)}\right)}{d^}\\=&(\frac{-(1 + 0)}{(^ + 3)^{2}})\\=&\frac{-1}{(^ + 3)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{(^ + 3)^{2}}\right)}{d^}\\=&-(\frac{-2(1 + 0)}{(^ + 3)^{3}})\\=&\frac{2}{(^ + 3)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{(^ + 3)^{3}}\right)}{d^}\\=&2(\frac{-3(1 + 0)}{(^ + 3)^{4}})\\=&\frac{-6}{(^ + 3)^{4}}\\ \end{split}\end{equation} \]