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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {x}^{ln(2 + x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {x}^{ln(x + 2)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {x}^{ln(x + 2)}\right)}{dx}\\=&({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))\\=&\frac{{x}^{ln(x + 2)}ln(x)}{(x + 2)} + \frac{{x}^{ln(x + 2)}ln(x + 2)}{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{x}^{ln(x + 2)}ln(x)}{(x + 2)} + \frac{{x}^{ln(x + 2)}ln(x + 2)}{x}\right)}{dx}\\=&(\frac{-(1 + 0)}{(x + 2)^{2}}){x}^{ln(x + 2)}ln(x) + \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x)}{(x + 2)} + \frac{{x}^{ln(x + 2)}}{(x + 2)(x)} + \frac{-{x}^{ln(x + 2)}ln(x + 2)}{x^{2}} + \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x + 2)}{x} + \frac{{x}^{ln(x + 2)}(1 + 0)}{x(x + 2)}\\=&\frac{-{x}^{ln(x + 2)}ln(x)}{(x + 2)^{2}} + \frac{{x}^{ln(x + 2)}ln^{2}(x)}{(x + 2)^{2}} + \frac{{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)x} + \frac{{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)x} - \frac{{x}^{ln(x + 2)}ln(x + 2)}{x^{2}} + \frac{2{x}^{ln(x + 2)}}{(x + 2)x} + \frac{{x}^{ln(x + 2)}ln^{2}(x + 2)}{x^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-{x}^{ln(x + 2)}ln(x)}{(x + 2)^{2}} + \frac{{x}^{ln(x + 2)}ln^{2}(x)}{(x + 2)^{2}} + \frac{{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)x} + \frac{{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)x} - \frac{{x}^{ln(x + 2)}ln(x + 2)}{x^{2}} + \frac{2{x}^{ln(x + 2)}}{(x + 2)x} + \frac{{x}^{ln(x + 2)}ln^{2}(x + 2)}{x^{2}}\right)}{dx}\\=&-(\frac{-2(1 + 0)}{(x + 2)^{3}}){x}^{ln(x + 2)}ln(x) - \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x)}{(x + 2)^{2}} - \frac{{x}^{ln(x + 2)}}{(x + 2)^{2}(x)} + (\frac{-2(1 + 0)}{(x + 2)^{3}}){x}^{ln(x + 2)}ln^{2}(x) + \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln^{2}(x)}{(x + 2)^{2}} + \frac{{x}^{ln(x + 2)}*2ln(x)}{(x + 2)^{2}(x)} + \frac{(\frac{-(1 + 0)}{(x + 2)^{2}}){x}^{ln(x + 2)}ln(x + 2)ln(x)}{x} + \frac{-{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)x^{2}} + \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x + 2)ln(x)}{(x + 2)x} + \frac{{x}^{ln(x + 2)}(1 + 0)ln(x)}{(x + 2)x(x + 2)} + \frac{{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)x(x)} + \frac{(\frac{-(1 + 0)}{(x + 2)^{2}}){x}^{ln(x + 2)}ln(x)ln(x + 2)}{x} + \frac{-{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)x^{2}} + \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x)ln(x + 2)}{(x + 2)x} + \frac{{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)x(x)} + \frac{{x}^{ln(x + 2)}ln(x)(1 + 0)}{(x + 2)x(x + 2)} - \frac{-2{x}^{ln(x + 2)}ln(x + 2)}{x^{3}} - \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x + 2)}{x^{2}} - \frac{{x}^{ln(x + 2)}(1 + 0)}{x^{2}(x + 2)} + \frac{2(\frac{-(1 + 0)}{(x + 2)^{2}}){x}^{ln(x + 2)}}{x} + \frac{2*-{x}^{ln(x + 2)}}{(x + 2)x^{2}} + \frac{2({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))}{(x + 2)x} + \frac{-2{x}^{ln(x + 2)}ln^{2}(x + 2)}{x^{3}} + \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln^{2}(x + 2)}{x^{2}} + \frac{{x}^{ln(x + 2)}*2ln(x + 2)(1 + 0)}{x^{2}(x + 2)}\\=&\frac{2{x}^{ln(x + 2)}ln(x)}{(x + 2)^{3}} - \frac{3{x}^{ln(x + 2)}ln^{2}(x)}{(x + 2)^{3}} + \frac{2{x}^{ln(x + 2)}ln^{2}(x)ln(x + 2)}{(x + 2)^{2}x} - \frac{2{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)^{2}x} + \frac{{x}^{ln(x + 2)}ln^{3}(x)}{(x + 2)^{3}} - \frac{{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)^{2}x} - \frac{{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)x^{2}} + \frac{2{x}^{ln(x + 2)}ln^{2}(x + 2)ln(x)}{(x + 2)x^{2}} - \frac{2{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)x^{2}} + \frac{{x}^{ln(x + 2)}ln(x + 2)ln^{2}(x)}{(x + 2)^{2}x} + \frac{{x}^{ln(x + 2)}ln(x)ln^{2}(x + 2)}{(x + 2)x^{2}} + \frac{6{x}^{ln(x + 2)}ln(x)}{(x + 2)^{2}x} + \frac{2{x}^{ln(x + 2)}ln(x + 2)}{x^{3}} - \frac{3{x}^{ln(x + 2)}ln^{2}(x + 2)}{x^{3}} + \frac{6{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)x^{2}} - \frac{3{x}^{ln(x + 2)}}{(x + 2)^{2}x} - \frac{3{x}^{ln(x + 2)}}{(x + 2)x^{2}} + \frac{{x}^{ln(x + 2)}ln^{3}(x + 2)}{x^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2{x}^{ln(x + 2)}ln(x)}{(x + 2)^{3}} - \frac{3{x}^{ln(x + 2)}ln^{2}(x)}{(x + 2)^{3}} + \frac{2{x}^{ln(x + 2)}ln^{2}(x)ln(x + 2)}{(x + 2)^{2}x} - \frac{2{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)^{2}x} + \frac{{x}^{ln(x + 2)}ln^{3}(x)}{(x + 2)^{3}} - \frac{{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)^{2}x} - \frac{{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)x^{2}} + \frac{2{x}^{ln(x + 2)}ln^{2}(x + 2)ln(x)}{(x + 2)x^{2}} - \frac{2{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)x^{2}} + \frac{{x}^{ln(x + 2)}ln(x + 2)ln^{2}(x)}{(x + 2)^{2}x} + \frac{{x}^{ln(x + 2)}ln(x)ln^{2}(x + 2)}{(x + 2)x^{2}} + \frac{6{x}^{ln(x + 2)}ln(x)}{(x + 2)^{2}x} + \frac{2{x}^{ln(x + 2)}ln(x + 2)}{x^{3}} - \frac{3{x}^{ln(x + 2)}ln^{2}(x + 2)}{x^{3}} + \frac{6{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)x^{2}} - \frac{3{x}^{ln(x + 2)}}{(x + 2)^{2}x} - \frac{3{x}^{ln(x + 2)}}{(x + 2)x^{2}} + \frac{{x}^{ln(x + 2)}ln^{3}(x + 2)}{x^{3}}\right)}{dx}\\=&2(\frac{-3(1 + 0)}{(x + 2)^{4}}){x}^{ln(x + 2)}ln(x) + \frac{2({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x)}{(x + 2)^{3}} + \frac{2{x}^{ln(x + 2)}}{(x + 2)^{3}(x)} - 3(\frac{-3(1 + 0)}{(x + 2)^{4}}){x}^{ln(x + 2)}ln^{2}(x) - \frac{3({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln^{2}(x)}{(x + 2)^{3}} - \frac{3{x}^{ln(x + 2)}*2ln(x)}{(x + 2)^{3}(x)} + \frac{2(\frac{-2(1 + 0)}{(x + 2)^{3}}){x}^{ln(x + 2)}ln^{2}(x)ln(x + 2)}{x} + \frac{2*-{x}^{ln(x + 2)}ln^{2}(x)ln(x + 2)}{(x + 2)^{2}x^{2}} + \frac{2({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln^{2}(x)ln(x + 2)}{(x + 2)^{2}x} + \frac{2{x}^{ln(x + 2)}*2ln(x)ln(x + 2)}{(x + 2)^{2}x(x)} + \frac{2{x}^{ln(x + 2)}ln^{2}(x)(1 + 0)}{(x + 2)^{2}x(x + 2)} - \frac{2(\frac{-2(1 + 0)}{(x + 2)^{3}}){x}^{ln(x + 2)}ln(x + 2)ln(x)}{x} - \frac{2*-{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)^{2}x^{2}} - \frac{2({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x + 2)ln(x)}{(x + 2)^{2}x} - \frac{2{x}^{ln(x + 2)}(1 + 0)ln(x)}{(x + 2)^{2}x(x + 2)} - \frac{2{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)^{2}x(x)} + (\frac{-3(1 + 0)}{(x + 2)^{4}}){x}^{ln(x + 2)}ln^{3}(x) + \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln^{3}(x)}{(x + 2)^{3}} + \frac{{x}^{ln(x + 2)}*3ln^{2}(x)}{(x + 2)^{3}(x)} - \frac{(\frac{-2(1 + 0)}{(x + 2)^{3}}){x}^{ln(x + 2)}ln(x)ln(x + 2)}{x} - \frac{-{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)^{2}x^{2}} - \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x)ln(x + 2)}{(x + 2)^{2}x} - \frac{{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)^{2}x(x)} - \frac{{x}^{ln(x + 2)}ln(x)(1 + 0)}{(x + 2)^{2}x(x + 2)} - \frac{(\frac{-(1 + 0)}{(x + 2)^{2}}){x}^{ln(x + 2)}ln(x + 2)ln(x)}{x^{2}} - \frac{-2{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)x^{3}} - \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x + 2)ln(x)}{(x + 2)x^{2}} - \frac{{x}^{ln(x + 2)}(1 + 0)ln(x)}{(x + 2)x^{2}(x + 2)} - \frac{{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)x^{2}(x)} + \frac{2(\frac{-(1 + 0)}{(x + 2)^{2}}){x}^{ln(x + 2)}ln^{2}(x + 2)ln(x)}{x^{2}} + \frac{2*-2{x}^{ln(x + 2)}ln^{2}(x + 2)ln(x)}{(x + 2)x^{3}} + \frac{2({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln^{2}(x + 2)ln(x)}{(x + 2)x^{2}} + \frac{2{x}^{ln(x + 2)}*2ln(x + 2)(1 + 0)ln(x)}{(x + 2)x^{2}(x + 2)} + \frac{2{x}^{ln(x + 2)}ln^{2}(x + 2)}{(x + 2)x^{2}(x)} - \frac{2(\frac{-(1 + 0)}{(x + 2)^{2}}){x}^{ln(x + 2)}ln(x)ln(x + 2)}{x^{2}} - \frac{2*-2{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)x^{3}} - \frac{2({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x)ln(x + 2)}{(x + 2)x^{2}} - \frac{2{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)x^{2}(x)} - \frac{2{x}^{ln(x + 2)}ln(x)(1 + 0)}{(x + 2)x^{2}(x + 2)} + \frac{(\frac{-2(1 + 0)}{(x + 2)^{3}}){x}^{ln(x + 2)}ln(x + 2)ln^{2}(x)}{x} + \frac{-{x}^{ln(x + 2)}ln(x + 2)ln^{2}(x)}{(x + 2)^{2}x^{2}} + \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x + 2)ln^{2}(x)}{(x + 2)^{2}x} + \frac{{x}^{ln(x + 2)}(1 + 0)ln^{2}(x)}{(x + 2)^{2}x(x + 2)} + \frac{{x}^{ln(x + 2)}ln(x + 2)*2ln(x)}{(x + 2)^{2}x(x)} + \frac{(\frac{-(1 + 0)}{(x + 2)^{2}}){x}^{ln(x + 2)}ln(x)ln^{2}(x + 2)}{x^{2}} + \frac{-2{x}^{ln(x + 2)}ln(x)ln^{2}(x + 2)}{(x + 2)x^{3}} + \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x)ln^{2}(x + 2)}{(x + 2)x^{2}} + \frac{{x}^{ln(x + 2)}ln^{2}(x + 2)}{(x + 2)x^{2}(x)} + \frac{{x}^{ln(x + 2)}ln(x)*2ln(x + 2)(1 + 0)}{(x + 2)x^{2}(x + 2)} + \frac{6(\frac{-2(1 + 0)}{(x + 2)^{3}}){x}^{ln(x + 2)}ln(x)}{x} + \frac{6*-{x}^{ln(x + 2)}ln(x)}{(x + 2)^{2}x^{2}} + \frac{6({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x)}{(x + 2)^{2}x} + \frac{6{x}^{ln(x + 2)}}{(x + 2)^{2}x(x)} + \frac{2*-3{x}^{ln(x + 2)}ln(x + 2)}{x^{4}} + \frac{2({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x + 2)}{x^{3}} + \frac{2{x}^{ln(x + 2)}(1 + 0)}{x^{3}(x + 2)} - \frac{3*-3{x}^{ln(x + 2)}ln^{2}(x + 2)}{x^{4}} - \frac{3({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln^{2}(x + 2)}{x^{3}} - \frac{3{x}^{ln(x + 2)}*2ln(x + 2)(1 + 0)}{x^{3}(x + 2)} + \frac{6(\frac{-(1 + 0)}{(x + 2)^{2}}){x}^{ln(x + 2)}ln(x + 2)}{x^{2}} + \frac{6*-2{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)x^{3}} + \frac{6({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln(x + 2)}{(x + 2)x^{2}} + \frac{6{x}^{ln(x + 2)}(1 + 0)}{(x + 2)x^{2}(x + 2)} - \frac{3(\frac{-2(1 + 0)}{(x + 2)^{3}}){x}^{ln(x + 2)}}{x} - \frac{3*-{x}^{ln(x + 2)}}{(x + 2)^{2}x^{2}} - \frac{3({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))}{(x + 2)^{2}x} - \frac{3(\frac{-(1 + 0)}{(x + 2)^{2}}){x}^{ln(x + 2)}}{x^{2}} - \frac{3*-2{x}^{ln(x + 2)}}{(x + 2)x^{3}} - \frac{3({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))}{(x + 2)x^{2}} + \frac{-3{x}^{ln(x + 2)}ln^{3}(x + 2)}{x^{4}} + \frac{({x}^{ln(x + 2)}((\frac{(1 + 0)}{(x + 2)})ln(x) + \frac{(ln(x + 2))(1)}{(x)}))ln^{3}(x + 2)}{x^{3}} + \frac{{x}^{ln(x + 2)}*3ln^{2}(x + 2)(1 + 0)}{x^{3}(x + 2)}\\=&\frac{-6{x}^{ln(x + 2)}ln(x)}{(x + 2)^{4}} + \frac{11{x}^{ln(x + 2)}ln^{2}(x)}{(x + 2)^{4}} + \frac{3{x}^{ln(x + 2)}ln^{3}(x)ln(x + 2)}{(x + 2)^{3}x} - \frac{7{x}^{ln(x + 2)}ln^{2}(x)ln(x + 2)}{(x + 2)^{3}x} - \frac{6{x}^{ln(x + 2)}ln^{3}(x)}{(x + 2)^{4}} + \frac{6{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)^{3}x} + \frac{2{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)^{3}x} - \frac{5{x}^{ln(x + 2)}ln^{2}(x)ln(x + 2)}{(x + 2)^{2}x^{2}} + \frac{3{x}^{ln(x + 2)}ln^{2}(x + 2)ln^{2}(x)}{(x + 2)^{2}x^{2}} + \frac{17{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)^{2}x^{2}} - \frac{5{x}^{ln(x + 2)}ln(x + 2)ln^{2}(x)}{(x + 2)^{3}x} + \frac{13{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)^{2}x^{2}} - \frac{5{x}^{ln(x + 2)}ln^{2}(x + 2)ln(x)}{(x + 2)^{2}x^{2}} + \frac{{x}^{ln(x + 2)}ln(x + 2)ln^{3}(x)}{(x + 2)^{3}x} + \frac{{x}^{ln(x + 2)}ln^{4}(x)}{(x + 2)^{4}} - \frac{{x}^{ln(x + 2)}ln(x)ln^{2}(x + 2)}{(x + 2)^{2}x^{2}} + \frac{2{x}^{ln(x + 2)}ln(x + 2)ln(x)}{(x + 2)x^{3}} - \frac{7{x}^{ln(x + 2)}ln^{2}(x + 2)ln(x)}{(x + 2)x^{3}} - \frac{{x}^{ln(x + 2)}ln(x + 2)ln^{2}(x)}{(x + 2)^{2}x^{2}} + \frac{6{x}^{ln(x + 2)}ln(x)ln(x + 2)}{(x + 2)x^{3}} + \frac{3{x}^{ln(x + 2)}ln^{2}(x)ln^{2}(x + 2)}{(x + 2)^{2}x^{2}} + \frac{3{x}^{ln(x + 2)}ln^{3}(x + 2)ln(x)}{(x + 2)x^{3}} - \frac{5{x}^{ln(x + 2)}ln(x)ln^{2}(x + 2)}{(x + 2)x^{3}} + \frac{{x}^{ln(x + 2)}ln(x)ln^{3}(x + 2)}{(x + 2)x^{3}} + \frac{12{x}^{ln(x + 2)}ln^{2}(x)}{(x + 2)^{3}x} - \frac{24{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)x^{3}} - \frac{12{x}^{ln(x + 2)}ln(x + 2)}{(x + 2)^{2}x^{2}} - \frac{12{x}^{ln(x + 2)}ln(x)}{(x + 2)^{2}x^{2}} + \frac{12{x}^{ln(x + 2)}ln^{2}(x + 2)}{(x + 2)x^{3}} - \frac{24{x}^{ln(x + 2)}ln(x)}{(x + 2)^{3}x} - \frac{6{x}^{ln(x + 2)}ln(x + 2)}{x^{4}} + \frac{11{x}^{ln(x + 2)}ln^{2}(x + 2)}{x^{4}} - \frac{6{x}^{ln(x + 2)}ln^{3}(x + 2)}{x^{4}} + \frac{8{x}^{ln(x + 2)}}{(x + 2)x^{3}} + \frac{8{x}^{ln(x + 2)}}{(x + 2)^{3}x} + \frac{18{x}^{ln(x + 2)}}{(x + 2)^{2}x^{2}} + \frac{{x}^{ln(x + 2)}ln^{4}(x + 2)}{x^{4}}\\ \end{split}\end{equation} \]