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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {(3 + x)}^{sin(x)} - {3}^{sin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (x + 3)^{sin(x)} - {3}^{sin(x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (x + 3)^{sin(x)} - {3}^{sin(x)}\right)}{dx}\\=&((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)})) - ({3}^{sin(x)}((cos(x))ln(3) + \frac{(sin(x))(0)}{(3)}))\\=&(x + 3)^{sin(x)}ln(x + 3)cos(x) + \frac{(x + 3)^{sin(x)}sin(x)}{(x + 3)} - {3}^{sin(x)}ln(3)cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( (x + 3)^{sin(x)}ln(x + 3)cos(x) + \frac{(x + 3)^{sin(x)}sin(x)}{(x + 3)} - {3}^{sin(x)}ln(3)cos(x)\right)}{dx}\\=&((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln(x + 3)cos(x) + \frac{(x + 3)^{sin(x)}(1 + 0)cos(x)}{(x + 3)} + (x + 3)^{sin(x)}ln(x + 3)*-sin(x) + (\frac{-(1 + 0)}{(x + 3)^{2}})(x + 3)^{sin(x)}sin(x) + \frac{((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))sin(x)}{(x + 3)} + \frac{(x + 3)^{sin(x)}cos(x)}{(x + 3)} - ({3}^{sin(x)}((cos(x))ln(3) + \frac{(sin(x))(0)}{(3)}))ln(3)cos(x) - \frac{{3}^{sin(x)}*0cos(x)}{(3)} - {3}^{sin(x)}ln(3)*-sin(x)\\=&(x + 3)^{sin(x)}ln^{2}(x + 3)cos^{2}(x) + \frac{2(x + 3)^{sin(x)}ln(x + 3)sin(x)cos(x)}{(x + 3)} + \frac{2(x + 3)^{sin(x)}cos(x)}{(x + 3)} - (x + 3)^{sin(x)}ln(x + 3)sin(x) - \frac{(x + 3)^{sin(x)}sin(x)}{(x + 3)^{2}} + \frac{(x + 3)^{sin(x)}sin^{2}(x)}{(x + 3)^{2}} - {3}^{sin(x)}ln^{2}(3)cos^{2}(x) + {3}^{sin(x)}ln(3)sin(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( (x + 3)^{sin(x)}ln^{2}(x + 3)cos^{2}(x) + \frac{2(x + 3)^{sin(x)}ln(x + 3)sin(x)cos(x)}{(x + 3)} + \frac{2(x + 3)^{sin(x)}cos(x)}{(x + 3)} - (x + 3)^{sin(x)}ln(x + 3)sin(x) - \frac{(x + 3)^{sin(x)}sin(x)}{(x + 3)^{2}} + \frac{(x + 3)^{sin(x)}sin^{2}(x)}{(x + 3)^{2}} - {3}^{sin(x)}ln^{2}(3)cos^{2}(x) + {3}^{sin(x)}ln(3)sin(x)\right)}{dx}\\=&((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln^{2}(x + 3)cos^{2}(x) + \frac{(x + 3)^{sin(x)}*2ln(x + 3)(1 + 0)cos^{2}(x)}{(x + 3)} + (x + 3)^{sin(x)}ln^{2}(x + 3)*-2cos(x)sin(x) + 2(\frac{-(1 + 0)}{(x + 3)^{2}})(x + 3)^{sin(x)}ln(x + 3)sin(x)cos(x) + \frac{2((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln(x + 3)sin(x)cos(x)}{(x + 3)} + \frac{2(x + 3)^{sin(x)}(1 + 0)sin(x)cos(x)}{(x + 3)(x + 3)} + \frac{2(x + 3)^{sin(x)}ln(x + 3)cos(x)cos(x)}{(x + 3)} + \frac{2(x + 3)^{sin(x)}ln(x + 3)sin(x)*-sin(x)}{(x + 3)} + 2(\frac{-(1 + 0)}{(x + 3)^{2}})(x + 3)^{sin(x)}cos(x) + \frac{2((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))cos(x)}{(x + 3)} + \frac{2(x + 3)^{sin(x)}*-sin(x)}{(x + 3)} - ((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln(x + 3)sin(x) - \frac{(x + 3)^{sin(x)}(1 + 0)sin(x)}{(x + 3)} - (x + 3)^{sin(x)}ln(x + 3)cos(x) - (\frac{-2(1 + 0)}{(x + 3)^{3}})(x + 3)^{sin(x)}sin(x) - \frac{((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))sin(x)}{(x + 3)^{2}} - \frac{(x + 3)^{sin(x)}cos(x)}{(x + 3)^{2}} + (\frac{-2(1 + 0)}{(x + 3)^{3}})(x + 3)^{sin(x)}sin^{2}(x) + \frac{((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))sin^{2}(x)}{(x + 3)^{2}} + \frac{(x + 3)^{sin(x)}*2sin(x)cos(x)}{(x + 3)^{2}} - ({3}^{sin(x)}((cos(x))ln(3) + \frac{(sin(x))(0)}{(3)}))ln^{2}(3)cos^{2}(x) - \frac{{3}^{sin(x)}*2ln(3)*0cos^{2}(x)}{(3)} - {3}^{sin(x)}ln^{2}(3)*-2cos(x)sin(x) + ({3}^{sin(x)}((cos(x))ln(3) + \frac{(sin(x))(0)}{(3)}))ln(3)sin(x) + \frac{{3}^{sin(x)}*0sin(x)}{(3)} + {3}^{sin(x)}ln(3)cos(x)\\=&(x + 3)^{sin(x)}ln^{3}(x + 3)cos^{3}(x) + \frac{3(x + 3)^{sin(x)}ln^{2}(x + 3)sin(x)cos^{2}(x)}{(x + 3)} + \frac{6(x + 3)^{sin(x)}ln(x + 3)cos^{2}(x)}{(x + 3)} - 3(x + 3)^{sin(x)}ln^{2}(x + 3)sin(x)cos(x) - \frac{3(x + 3)^{sin(x)}ln(x + 3)sin(x)cos(x)}{(x + 3)^{2}} + \frac{3(x + 3)^{sin(x)}ln(x + 3)sin^{2}(x)cos(x)}{(x + 3)^{2}} + \frac{6(x + 3)^{sin(x)}sin(x)cos(x)}{(x + 3)^{2}} - \frac{3(x + 3)^{sin(x)}ln(x + 3)sin^{2}(x)}{(x + 3)} - \frac{3(x + 3)^{sin(x)}cos(x)}{(x + 3)^{2}} - \frac{3(x + 3)^{sin(x)}sin(x)}{(x + 3)} - (x + 3)^{sin(x)}ln(x + 3)cos(x) + \frac{2(x + 3)^{sin(x)}sin(x)}{(x + 3)^{3}} - \frac{3(x + 3)^{sin(x)}sin^{2}(x)}{(x + 3)^{3}} + \frac{(x + 3)^{sin(x)}sin^{3}(x)}{(x + 3)^{3}} - {3}^{sin(x)}ln^{3}(3)cos^{3}(x) + 3 * {3}^{sin(x)}ln^{2}(3)sin(x)cos(x) + {3}^{sin(x)}ln(3)cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( (x + 3)^{sin(x)}ln^{3}(x + 3)cos^{3}(x) + \frac{3(x + 3)^{sin(x)}ln^{2}(x + 3)sin(x)cos^{2}(x)}{(x + 3)} + \frac{6(x + 3)^{sin(x)}ln(x + 3)cos^{2}(x)}{(x + 3)} - 3(x + 3)^{sin(x)}ln^{2}(x + 3)sin(x)cos(x) - \frac{3(x + 3)^{sin(x)}ln(x + 3)sin(x)cos(x)}{(x + 3)^{2}} + \frac{3(x + 3)^{sin(x)}ln(x + 3)sin^{2}(x)cos(x)}{(x + 3)^{2}} + \frac{6(x + 3)^{sin(x)}sin(x)cos(x)}{(x + 3)^{2}} - \frac{3(x + 3)^{sin(x)}ln(x + 3)sin^{2}(x)}{(x + 3)} - \frac{3(x + 3)^{sin(x)}cos(x)}{(x + 3)^{2}} - \frac{3(x + 3)^{sin(x)}sin(x)}{(x + 3)} - (x + 3)^{sin(x)}ln(x + 3)cos(x) + \frac{2(x + 3)^{sin(x)}sin(x)}{(x + 3)^{3}} - \frac{3(x + 3)^{sin(x)}sin^{2}(x)}{(x + 3)^{3}} + \frac{(x + 3)^{sin(x)}sin^{3}(x)}{(x + 3)^{3}} - {3}^{sin(x)}ln^{3}(3)cos^{3}(x) + 3 * {3}^{sin(x)}ln^{2}(3)sin(x)cos(x) + {3}^{sin(x)}ln(3)cos(x)\right)}{dx}\\=&((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln^{3}(x + 3)cos^{3}(x) + \frac{(x + 3)^{sin(x)}*3ln^{2}(x + 3)(1 + 0)cos^{3}(x)}{(x + 3)} + (x + 3)^{sin(x)}ln^{3}(x + 3)*-3cos^{2}(x)sin(x) + 3(\frac{-(1 + 0)}{(x + 3)^{2}})(x + 3)^{sin(x)}ln^{2}(x + 3)sin(x)cos^{2}(x) + \frac{3((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln^{2}(x + 3)sin(x)cos^{2}(x)}{(x + 3)} + \frac{3(x + 3)^{sin(x)}*2ln(x + 3)(1 + 0)sin(x)cos^{2}(x)}{(x + 3)(x + 3)} + \frac{3(x + 3)^{sin(x)}ln^{2}(x + 3)cos(x)cos^{2}(x)}{(x + 3)} + \frac{3(x + 3)^{sin(x)}ln^{2}(x + 3)sin(x)*-2cos(x)sin(x)}{(x + 3)} + 6(\frac{-(1 + 0)}{(x + 3)^{2}})(x + 3)^{sin(x)}ln(x + 3)cos^{2}(x) + \frac{6((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln(x + 3)cos^{2}(x)}{(x + 3)} + \frac{6(x + 3)^{sin(x)}(1 + 0)cos^{2}(x)}{(x + 3)(x + 3)} + \frac{6(x + 3)^{sin(x)}ln(x + 3)*-2cos(x)sin(x)}{(x + 3)} - 3((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln^{2}(x + 3)sin(x)cos(x) - \frac{3(x + 3)^{sin(x)}*2ln(x + 3)(1 + 0)sin(x)cos(x)}{(x + 3)} - 3(x + 3)^{sin(x)}ln^{2}(x + 3)cos(x)cos(x) - 3(x + 3)^{sin(x)}ln^{2}(x + 3)sin(x)*-sin(x) - 3(\frac{-2(1 + 0)}{(x + 3)^{3}})(x + 3)^{sin(x)}ln(x + 3)sin(x)cos(x) - \frac{3((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln(x + 3)sin(x)cos(x)}{(x + 3)^{2}} - \frac{3(x + 3)^{sin(x)}(1 + 0)sin(x)cos(x)}{(x + 3)^{2}(x + 3)} - \frac{3(x + 3)^{sin(x)}ln(x + 3)cos(x)cos(x)}{(x + 3)^{2}} - \frac{3(x + 3)^{sin(x)}ln(x + 3)sin(x)*-sin(x)}{(x + 3)^{2}} + 3(\frac{-2(1 + 0)}{(x + 3)^{3}})(x + 3)^{sin(x)}ln(x + 3)sin^{2}(x)cos(x) + \frac{3((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln(x + 3)sin^{2}(x)cos(x)}{(x + 3)^{2}} + \frac{3(x + 3)^{sin(x)}(1 + 0)sin^{2}(x)cos(x)}{(x + 3)^{2}(x + 3)} + \frac{3(x + 3)^{sin(x)}ln(x + 3)*2sin(x)cos(x)cos(x)}{(x + 3)^{2}} + \frac{3(x + 3)^{sin(x)}ln(x + 3)sin^{2}(x)*-sin(x)}{(x + 3)^{2}} + 6(\frac{-2(1 + 0)}{(x + 3)^{3}})(x + 3)^{sin(x)}sin(x)cos(x) + \frac{6((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))sin(x)cos(x)}{(x + 3)^{2}} + \frac{6(x + 3)^{sin(x)}cos(x)cos(x)}{(x + 3)^{2}} + \frac{6(x + 3)^{sin(x)}sin(x)*-sin(x)}{(x + 3)^{2}} - 3(\frac{-(1 + 0)}{(x + 3)^{2}})(x + 3)^{sin(x)}ln(x + 3)sin^{2}(x) - \frac{3((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln(x + 3)sin^{2}(x)}{(x + 3)} - \frac{3(x + 3)^{sin(x)}(1 + 0)sin^{2}(x)}{(x + 3)(x + 3)} - \frac{3(x + 3)^{sin(x)}ln(x + 3)*2sin(x)cos(x)}{(x + 3)} - 3(\frac{-2(1 + 0)}{(x + 3)^{3}})(x + 3)^{sin(x)}cos(x) - \frac{3((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))cos(x)}{(x + 3)^{2}} - \frac{3(x + 3)^{sin(x)}*-sin(x)}{(x + 3)^{2}} - 3(\frac{-(1 + 0)}{(x + 3)^{2}})(x + 3)^{sin(x)}sin(x) - \frac{3((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))sin(x)}{(x + 3)} - \frac{3(x + 3)^{sin(x)}cos(x)}{(x + 3)} - ((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))ln(x + 3)cos(x) - \frac{(x + 3)^{sin(x)}(1 + 0)cos(x)}{(x + 3)} - (x + 3)^{sin(x)}ln(x + 3)*-sin(x) + 2(\frac{-3(1 + 0)}{(x + 3)^{4}})(x + 3)^{sin(x)}sin(x) + \frac{2((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))sin(x)}{(x + 3)^{3}} + \frac{2(x + 3)^{sin(x)}cos(x)}{(x + 3)^{3}} - 3(\frac{-3(1 + 0)}{(x + 3)^{4}})(x + 3)^{sin(x)}sin^{2}(x) - \frac{3((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))sin^{2}(x)}{(x + 3)^{3}} - \frac{3(x + 3)^{sin(x)}*2sin(x)cos(x)}{(x + 3)^{3}} + (\frac{-3(1 + 0)}{(x + 3)^{4}})(x + 3)^{sin(x)}sin^{3}(x) + \frac{((x + 3)^{sin(x)}((cos(x))ln(x + 3) + \frac{(sin(x))(1 + 0)}{(x + 3)}))sin^{3}(x)}{(x + 3)^{3}} + \frac{(x + 3)^{sin(x)}*3sin^{2}(x)cos(x)}{(x + 3)^{3}} - ({3}^{sin(x)}((cos(x))ln(3) + \frac{(sin(x))(0)}{(3)}))ln^{3}(3)cos^{3}(x) - \frac{{3}^{sin(x)}*3ln^{2}(3)*0cos^{3}(x)}{(3)} - {3}^{sin(x)}ln^{3}(3)*-3cos^{2}(x)sin(x) + 3({3}^{sin(x)}((cos(x))ln(3) + \frac{(sin(x))(0)}{(3)}))ln^{2}(3)sin(x)cos(x) + \frac{3 * {3}^{sin(x)}*2ln(3)*0sin(x)cos(x)}{(3)} + 3 * {3}^{sin(x)}ln^{2}(3)cos(x)cos(x) + 3 * {3}^{sin(x)}ln^{2}(3)sin(x)*-sin(x) + ({3}^{sin(x)}((cos(x))ln(3) + \frac{(sin(x))(0)}{(3)}))ln(3)cos(x) + \frac{{3}^{sin(x)}*0cos(x)}{(3)} + {3}^{sin(x)}ln(3)*-sin(x)\\=&(x + 3)^{sin(x)}ln^{4}(x + 3)cos^{4}(x) + \frac{4(x + 3)^{sin(x)}ln^{3}(x + 3)sin(x)cos^{3}(x)}{(x + 3)} + \frac{12(x + 3)^{sin(x)}ln^{2}(x + 3)cos^{3}(x)}{(x + 3)} - 6(x + 3)^{sin(x)}ln^{3}(x + 3)sin(x)cos^{2}(x) + \frac{24(x + 3)^{sin(x)}ln(x + 3)sin(x)cos^{2}(x)}{(x + 3)^{2}} + \frac{6(x + 3)^{sin(x)}ln^{2}(x + 3)sin^{2}(x)cos^{2}(x)}{(x + 3)^{2}} - \frac{12(x + 3)^{sin(x)}ln^{2}(x + 3)sin^{2}(x)cos(x)}{(x + 3)} - \frac{12(x + 3)^{sin(x)}ln(x + 3)cos^{2}(x)}{(x + 3)^{2}} + \frac{12(x + 3)^{sin(x)}cos^{2}(x)}{(x + 3)^{2}} - \frac{28(x + 3)^{sin(x)}ln(x + 3)sin(x)cos(x)}{(x + 3)} - 4(x + 3)^{sin(x)}ln^{2}(x + 3)cos^{2}(x) + 6 * {3}^{sin(x)}ln^{3}(3)sin(x)cos^{2}(x) + \frac{8(x + 3)^{sin(x)}ln(x + 3)sin(x)cos(x)}{(x + 3)^{3}} - \frac{6(x + 3)^{sin(x)}ln^{2}(x + 3)sin(x)cos^{2}(x)}{(x + 3)^{2}} - \frac{12(x + 3)^{sin(x)}ln(x + 3)sin^{2}(x)cos(x)}{(x + 3)^{3}} - \frac{24(x + 3)^{sin(x)}sin(x)cos(x)}{(x + 3)^{3}} + \frac{4(x + 3)^{sin(x)}ln(x + 3)sin^{3}(x)cos(x)}{(x + 3)^{3}} + \frac{12(x + 3)^{sin(x)}sin^{2}(x)cos(x)}{(x + 3)^{3}} + \frac{6(x + 3)^{sin(x)}ln(x + 3)sin^{2}(x)}{(x + 3)^{2}} - \frac{6(x + 3)^{sin(x)}ln(x + 3)sin^{3}(x)}{(x + 3)^{2}} - \frac{12(x + 3)^{sin(x)}sin^{2}(x)}{(x + 3)^{2}} + \frac{8(x + 3)^{sin(x)}cos(x)}{(x + 3)^{3}} - \frac{4(x + 3)^{sin(x)}cos(x)}{(x + 3)} + 3(x + 3)^{sin(x)}ln^{2}(x + 3)sin^{2}(x) - \frac{6(x + 3)^{sin(x)}sin(x)}{(x + 3)^{4}} + \frac{11(x + 3)^{sin(x)}sin^{2}(x)}{(x + 3)^{4}} - \frac{6(x + 3)^{sin(x)}sin^{3}(x)}{(x + 3)^{4}} + \frac{6(x + 3)^{sin(x)}sin(x)}{(x + 3)^{2}} + \frac{(x + 3)^{sin(x)}sin^{4}(x)}{(x + 3)^{4}} - {3}^{sin(x)}ln^{4}(3)cos^{4}(x) - 3 * {3}^{sin(x)}ln^{2}(3)sin^{2}(x) + 4 * {3}^{sin(x)}ln^{2}(3)cos^{2}(x) + (x + 3)^{sin(x)}ln(x + 3)sin(x) - {3}^{sin(x)}ln(3)sin(x)\\ \end{split}\end{equation} \]