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 7 questions in this calculation: for each question, the 2 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/7]Find\ the\ second\ derivative\ of\ function\ xx\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = x^{2}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( x^{2}\right)}{dx}\\=&2x\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2x\right)}{dx}\\=&2\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/7]Find\ the\ second\ derivative\ of\ function\ xsin(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xsin(x)\right)}{dx}\\=&sin(x) + xcos(x)\\=&sin(x) + xcos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sin(x) + xcos(x)\right)}{dx}\\=&cos(x) + cos(x) + x*-sin(x)\\=&2cos(x) - xsin(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/7]Find\ the\ second\ derivative\ of\ function\ xlog_{x}^{xx}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = xlog_{x}^{x^{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xlog_{x}^{x^{2}}\right)}{dx}\\=&log_{x}^{x^{2}} + x(\frac{(\frac{(2x)}{(x^{2})} - \frac{(1)log_{x}^{x^{2}}}{(x)})}{(ln(x))})\\=& - \frac{log_{x}^{x^{2}}}{ln(x)} + \frac{2}{ln(x)} + log_{x}^{x^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{log_{x}^{x^{2}}}{ln(x)} + \frac{2}{ln(x)} + log_{x}^{x^{2}}\right)}{dx}\\=& - \frac{(\frac{(\frac{(2x)}{(x^{2})} - \frac{(1)log_{x}^{x^{2}}}{(x)})}{(ln(x))})}{ln(x)} - \frac{log_{x}^{x^{2}}*-1}{ln^{2}(x)(x)} + \frac{2*-1}{ln^{2}(x)(x)} + (\frac{(\frac{(2x)}{(x^{2})} - \frac{(1)log_{x}^{x^{2}}}{(x)})}{(ln(x))})\\=& - \frac{4}{xln^{2}(x)} + \frac{2log_{x}^{x^{2}}}{xln^{2}(x)} + \frac{2}{xln(x)} - \frac{log_{x}^{x^{2}}}{xln(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/7]Find\ the\ second\ derivative\ of\ function\ xsin(x)cos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xsin(x)cos(x)\right)}{dx}\\=&sin(x)cos(x) + xcos(x)cos(x) + xsin(x)*-sin(x)\\=&sin(x)cos(x) + xcos^{2}(x) - xsin^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sin(x)cos(x) + xcos^{2}(x) - xsin^{2}(x)\right)}{dx}\\=&cos(x)cos(x) + sin(x)*-sin(x) + cos^{2}(x) + x*-2cos(x)sin(x) - sin^{2}(x) - x*2sin(x)cos(x)\\=&2cos^{2}(x) - 2sin^{2}(x) - 4xsin(x)cos(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[5/7]Find\ the\ second\ derivative\ of\ function\ sin(cos(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(cos(x))\right)}{dx}\\=&cos(cos(x))*-sin(x)\\=&-sin(x)cos(cos(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(x)cos(cos(x))\right)}{dx}\\=&-cos(x)cos(cos(x)) - sin(x)*-sin(cos(x))*-sin(x)\\=&-cos(x)cos(cos(x)) - sin(cos(x))sin^{2}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[6/7]Find\ the\ second\ derivative\ of\ function\ cos(sin(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(sin(x))\right)}{dx}\\=&-sin(sin(x))cos(x)\\=&-sin(sin(x))cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(sin(x))cos(x)\right)}{dx}\\=&-cos(sin(x))cos(x)cos(x) - sin(sin(x))*-sin(x)\\=&-cos^{2}(x)cos(sin(x)) + sin(x)sin(sin(x))\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[7/7]Find\ the\ second\ derivative\ of\ function\ sin(ln(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(ln(x))\right)}{dx}\\=&\frac{cos(ln(x))}{(x)}\\=&\frac{cos(ln(x))}{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(ln(x))}{x}\right)}{dx}\\=&\frac{-cos(ln(x))}{x^{2}} + \frac{-sin(ln(x))}{x(x)}\\=&\frac{-cos(ln(x))}{x^{2}} - \frac{sin(ln(x))}{x^{2}}\\ \end{split}\end{equation} \]
>>注：本次最多计算 7 道题。
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/7]Find\ the\ second\ derivative\ of\ function\ xx\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = x^{2}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( x^{2}\right)}{dx}\\=&2x\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2x\right)}{dx}\\=&2\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/7]Find\ the\ second\ derivative\ of\ function\ xsin(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xsin(x)\right)}{dx}\\=&sin(x) + xcos(x)\\=&sin(x) + xcos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sin(x) + xcos(x)\right)}{dx}\\=&cos(x) + cos(x) + x*-sin(x)\\=&2cos(x) - xsin(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/7]Find\ the\ second\ derivative\ of\ function\ xlog_{x}^{xx}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = xlog_{x}^{x^{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xlog_{x}^{x^{2}}\right)}{dx}\\=&log_{x}^{x^{2}} + x(\frac{(\frac{(2x)}{(x^{2})} - \frac{(1)log_{x}^{x^{2}}}{(x)})}{(ln(x))})\\=& - \frac{log_{x}^{x^{2}}}{ln(x)} + \frac{2}{ln(x)} + log_{x}^{x^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{log_{x}^{x^{2}}}{ln(x)} + \frac{2}{ln(x)} + log_{x}^{x^{2}}\right)}{dx}\\=& - \frac{(\frac{(\frac{(2x)}{(x^{2})} - \frac{(1)log_{x}^{x^{2}}}{(x)})}{(ln(x))})}{ln(x)} - \frac{log_{x}^{x^{2}}*-1}{ln^{2}(x)(x)} + \frac{2*-1}{ln^{2}(x)(x)} + (\frac{(\frac{(2x)}{(x^{2})} - \frac{(1)log_{x}^{x^{2}}}{(x)})}{(ln(x))})\\=& - \frac{4}{xln^{2}(x)} + \frac{2log_{x}^{x^{2}}}{xln^{2}(x)} + \frac{2}{xln(x)} - \frac{log_{x}^{x^{2}}}{xln(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/7]Find\ the\ second\ derivative\ of\ function\ xsin(x)cos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xsin(x)cos(x)\right)}{dx}\\=&sin(x)cos(x) + xcos(x)cos(x) + xsin(x)*-sin(x)\\=&sin(x)cos(x) + xcos^{2}(x) - xsin^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sin(x)cos(x) + xcos^{2}(x) - xsin^{2}(x)\right)}{dx}\\=&cos(x)cos(x) + sin(x)*-sin(x) + cos^{2}(x) + x*-2cos(x)sin(x) - sin^{2}(x) - x*2sin(x)cos(x)\\=&2cos^{2}(x) - 2sin^{2}(x) - 4xsin(x)cos(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[5/7]Find\ the\ second\ derivative\ of\ function\ sin(cos(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(cos(x))\right)}{dx}\\=&cos(cos(x))*-sin(x)\\=&-sin(x)cos(cos(x))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(x)cos(cos(x))\right)}{dx}\\=&-cos(x)cos(cos(x)) - sin(x)*-sin(cos(x))*-sin(x)\\=&-cos(x)cos(cos(x)) - sin(cos(x))sin^{2}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[6/7]Find\ the\ second\ derivative\ of\ function\ cos(sin(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(sin(x))\right)}{dx}\\=&-sin(sin(x))cos(x)\\=&-sin(sin(x))cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(sin(x))cos(x)\right)}{dx}\\=&-cos(sin(x))cos(x)cos(x) - sin(sin(x))*-sin(x)\\=&-cos^{2}(x)cos(sin(x)) + sin(x)sin(sin(x))\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[7/7]Find\ the\ second\ derivative\ of\ function\ sin(ln(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(ln(x))\right)}{dx}\\=&\frac{cos(ln(x))}{(x)}\\=&\frac{cos(ln(x))}{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(ln(x))}{x}\right)}{dx}\\=&\frac{-cos(ln(x))}{x^{2}} + \frac{-sin(ln(x))}{x(x)}\\=&\frac{-cos(ln(x))}{x^{2}} - \frac{sin(ln(x))}{x^{2}}\\ \end{split}\end{equation} \]
>>注：本次最多计算 7 道题。