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 2 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ lg(x){10}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {10}^{x}lg(x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {10}^{x}lg(x)\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))lg(x) + \frac{{10}^{x}}{ln{10}(x)}\\=&{10}^{x}ln(10)lg(x) + \frac{{10}^{x}}{xln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln(10)lg(x) + \frac{{10}^{x}}{xln{10}}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)lg(x) + \frac{{10}^{x}*0lg(x)}{(10)} + \frac{{10}^{x}ln(10)}{ln{10}(x)} + \frac{-{10}^{x}}{x^{2}ln{10}} + \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{xln{10}} + \frac{{10}^{x}*-0}{xln^{2}{10}}\\=&{10}^{x}ln^{2}(10)lg(x) + \frac{{10}^{x}ln(10)}{xln{10}} + \frac{{10}^{x}ln(10)}{xln{10}} - \frac{{10}^{x}}{x^{2}ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln^{2}(10)lg(x) + \frac{{10}^{x}ln(10)}{xln{10}} + \frac{{10}^{x}ln(10)}{xln{10}} - \frac{{10}^{x}}{x^{2}ln{10}}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{2}(10)lg(x) + \frac{{10}^{x}*2ln(10)*0lg(x)}{(10)} + \frac{{10}^{x}ln^{2}(10)}{ln{10}(x)} + \frac{-{10}^{x}ln(10)}{x^{2}ln{10}} + \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{xln{10}} + \frac{{10}^{x}*-0ln(10)}{xln^{2}{10}} + \frac{{10}^{x}*0}{xln{10}(10)} + \frac{-{10}^{x}ln(10)}{x^{2}ln{10}} + \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{xln{10}} + \frac{{10}^{x}*0}{x(10)ln{10}} + \frac{{10}^{x}ln(10)*-0}{xln^{2}{10}} - \frac{-2 * {10}^{x}}{x^{3}ln{10}} - \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{2}ln{10}} - \frac{{10}^{x}*-0}{x^{2}ln^{2}{10}}\\=&{10}^{x}ln^{3}(10)lg(x) + \frac{{10}^{x}ln^{2}(10)}{xln{10}} - \frac{{10}^{x}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{x}ln^{2}(10)}{xln{10}} - \frac{2 * {10}^{x}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{x}}{x^{3}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln^{3}(10)lg(x) + \frac{{10}^{x}ln^{2}(10)}{xln{10}} - \frac{{10}^{x}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{x}ln^{2}(10)}{xln{10}} - \frac{2 * {10}^{x}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{x}}{x^{3}ln{10}}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{3}(10)lg(x) + \frac{{10}^{x}*3ln^{2}(10)*0lg(x)}{(10)} + \frac{{10}^{x}ln^{3}(10)}{ln{10}(x)} + \frac{-{10}^{x}ln^{2}(10)}{x^{2}ln{10}} + \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{2}(10)}{xln{10}} + \frac{{10}^{x}*-0ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{x}*2ln(10)*0}{xln{10}(10)} - \frac{-2 * {10}^{x}ln(10)}{x^{3}ln{10}} - \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{x^{2}ln{10}} - \frac{{10}^{x}*-0ln(10)}{x^{2}ln^{2}{10}} - \frac{{10}^{x}*0}{x^{2}ln{10}(10)} + \frac{2*-{10}^{x}ln^{2}(10)}{x^{2}ln{10}} + \frac{2({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{2}(10)}{xln{10}} + \frac{2 * {10}^{x}*2ln(10)*0}{x(10)ln{10}} + \frac{2 * {10}^{x}ln^{2}(10)*-0}{xln^{2}{10}} - \frac{2*-2 * {10}^{x}ln(10)}{x^{3}ln{10}} - \frac{2({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{x^{2}ln{10}} - \frac{2 * {10}^{x}*0}{x^{2}(10)ln{10}} - \frac{2 * {10}^{x}ln(10)*-0}{x^{2}ln^{2}{10}} + \frac{2*-3 * {10}^{x}}{x^{4}ln{10}} + \frac{2({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{3}ln{10}} + \frac{2 * {10}^{x}*-0}{x^{3}ln^{2}{10}}\\=&{10}^{x}ln^{4}(10)lg(x) + \frac{{10}^{x}ln^{3}(10)}{xln{10}} - \frac{{10}^{x}ln^{2}(10)}{x^{2}ln{10}} + \frac{3 * {10}^{x}ln^{3}(10)}{xln{10}} + \frac{2 * {10}^{x}ln(10)}{x^{3}ln{10}} - \frac{5 * {10}^{x}ln^{2}(10)}{x^{2}ln{10}} + \frac{6 * {10}^{x}ln(10)}{x^{3}ln{10}} - \frac{6 * {10}^{x}}{x^{4}ln{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ ln(x){10}^{x} + \frac{{10}^{x}}{(xln(10))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {10}^{x}ln(x) + \frac{{10}^{x}}{xln(10)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {10}^{x}ln(x) + \frac{{10}^{x}}{xln(10)}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(x) + \frac{{10}^{x}}{(x)} + \frac{-{10}^{x}}{x^{2}ln(10)} + \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{xln(10)} + \frac{{10}^{x}*-0}{xln^{2}(10)(10)}\\=&{10}^{x}ln(10)ln(x) - \frac{{10}^{x}}{x^{2}ln(10)} + \frac{2 * {10}^{x}}{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln(10)ln(x) - \frac{{10}^{x}}{x^{2}ln(10)} + \frac{2 * {10}^{x}}{x}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)ln(x) + \frac{{10}^{x}*0ln(x)}{(10)} + \frac{{10}^{x}ln(10)}{(x)} - \frac{-2 * {10}^{x}}{x^{3}ln(10)} - \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{2}ln(10)} - \frac{{10}^{x}*-0}{x^{2}ln^{2}(10)(10)} + \frac{2*-{10}^{x}}{x^{2}} + \frac{2({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x}\\=&{10}^{x}ln^{2}(10)ln(x) + \frac{3 * {10}^{x}ln(10)}{x} + \frac{2 * {10}^{x}}{x^{3}ln(10)} - \frac{3 * {10}^{x}}{x^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln^{2}(10)ln(x) + \frac{3 * {10}^{x}ln(10)}{x} + \frac{2 * {10}^{x}}{x^{3}ln(10)} - \frac{3 * {10}^{x}}{x^{2}}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{2}(10)ln(x) + \frac{{10}^{x}*2ln(10)*0ln(x)}{(10)} + \frac{{10}^{x}ln^{2}(10)}{(x)} + \frac{3*-{10}^{x}ln(10)}{x^{2}} + \frac{3({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{x} + \frac{3 * {10}^{x}*0}{x(10)} + \frac{2*-3 * {10}^{x}}{x^{4}ln(10)} + \frac{2({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{3}ln(10)} + \frac{2 * {10}^{x}*-0}{x^{3}ln^{2}(10)(10)} - \frac{3*-2 * {10}^{x}}{x^{3}} - \frac{3({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{2}}\\=&{10}^{x}ln^{3}(10)ln(x) + \frac{4 * {10}^{x}ln^{2}(10)}{x} - \frac{6 * {10}^{x}ln(10)}{x^{2}} - \frac{6 * {10}^{x}}{x^{4}ln(10)} + \frac{8 * {10}^{x}}{x^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln^{3}(10)ln(x) + \frac{4 * {10}^{x}ln^{2}(10)}{x} - \frac{6 * {10}^{x}ln(10)}{x^{2}} - \frac{6 * {10}^{x}}{x^{4}ln(10)} + \frac{8 * {10}^{x}}{x^{3}}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{3}(10)ln(x) + \frac{{10}^{x}*3ln^{2}(10)*0ln(x)}{(10)} + \frac{{10}^{x}ln^{3}(10)}{(x)} + \frac{4*-{10}^{x}ln^{2}(10)}{x^{2}} + \frac{4({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{2}(10)}{x} + \frac{4 * {10}^{x}*2ln(10)*0}{x(10)} - \frac{6*-2 * {10}^{x}ln(10)}{x^{3}} - \frac{6({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{x^{2}} - \frac{6 * {10}^{x}*0}{x^{2}(10)} - \frac{6*-4 * {10}^{x}}{x^{5}ln(10)} - \frac{6({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{4}ln(10)} - \frac{6 * {10}^{x}*-0}{x^{4}ln^{2}(10)(10)} + \frac{8*-3 * {10}^{x}}{x^{4}} + \frac{8({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{3}}\\=&{10}^{x}ln^{4}(10)ln(x) + \frac{5 * {10}^{x}ln^{3}(10)}{x} - \frac{10 * {10}^{x}ln^{2}(10)}{x^{2}} + \frac{20 * {10}^{x}ln(10)}{x^{3}} + \frac{24 * {10}^{x}}{x^{5}ln(10)} - \frac{30 * {10}^{x}}{x^{4}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/2]Find\ the\ 4th\ derivative\ of\ function\ lg(x){10}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {10}^{x}lg(x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {10}^{x}lg(x)\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))lg(x) + \frac{{10}^{x}}{ln{10}(x)}\\=&{10}^{x}ln(10)lg(x) + \frac{{10}^{x}}{xln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln(10)lg(x) + \frac{{10}^{x}}{xln{10}}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)lg(x) + \frac{{10}^{x}*0lg(x)}{(10)} + \frac{{10}^{x}ln(10)}{ln{10}(x)} + \frac{-{10}^{x}}{x^{2}ln{10}} + \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{xln{10}} + \frac{{10}^{x}*-0}{xln^{2}{10}}\\=&{10}^{x}ln^{2}(10)lg(x) + \frac{{10}^{x}ln(10)}{xln{10}} + \frac{{10}^{x}ln(10)}{xln{10}} - \frac{{10}^{x}}{x^{2}ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln^{2}(10)lg(x) + \frac{{10}^{x}ln(10)}{xln{10}} + \frac{{10}^{x}ln(10)}{xln{10}} - \frac{{10}^{x}}{x^{2}ln{10}}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{2}(10)lg(x) + \frac{{10}^{x}*2ln(10)*0lg(x)}{(10)} + \frac{{10}^{x}ln^{2}(10)}{ln{10}(x)} + \frac{-{10}^{x}ln(10)}{x^{2}ln{10}} + \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{xln{10}} + \frac{{10}^{x}*-0ln(10)}{xln^{2}{10}} + \frac{{10}^{x}*0}{xln{10}(10)} + \frac{-{10}^{x}ln(10)}{x^{2}ln{10}} + \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{xln{10}} + \frac{{10}^{x}*0}{x(10)ln{10}} + \frac{{10}^{x}ln(10)*-0}{xln^{2}{10}} - \frac{-2 * {10}^{x}}{x^{3}ln{10}} - \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{2}ln{10}} - \frac{{10}^{x}*-0}{x^{2}ln^{2}{10}}\\=&{10}^{x}ln^{3}(10)lg(x) + \frac{{10}^{x}ln^{2}(10)}{xln{10}} - \frac{{10}^{x}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{x}ln^{2}(10)}{xln{10}} - \frac{2 * {10}^{x}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{x}}{x^{3}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln^{3}(10)lg(x) + \frac{{10}^{x}ln^{2}(10)}{xln{10}} - \frac{{10}^{x}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{x}ln^{2}(10)}{xln{10}} - \frac{2 * {10}^{x}ln(10)}{x^{2}ln{10}} + \frac{2 * {10}^{x}}{x^{3}ln{10}}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{3}(10)lg(x) + \frac{{10}^{x}*3ln^{2}(10)*0lg(x)}{(10)} + \frac{{10}^{x}ln^{3}(10)}{ln{10}(x)} + \frac{-{10}^{x}ln^{2}(10)}{x^{2}ln{10}} + \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{2}(10)}{xln{10}} + \frac{{10}^{x}*-0ln^{2}(10)}{xln^{2}{10}} + \frac{{10}^{x}*2ln(10)*0}{xln{10}(10)} - \frac{-2 * {10}^{x}ln(10)}{x^{3}ln{10}} - \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{x^{2}ln{10}} - \frac{{10}^{x}*-0ln(10)}{x^{2}ln^{2}{10}} - \frac{{10}^{x}*0}{x^{2}ln{10}(10)} + \frac{2*-{10}^{x}ln^{2}(10)}{x^{2}ln{10}} + \frac{2({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{2}(10)}{xln{10}} + \frac{2 * {10}^{x}*2ln(10)*0}{x(10)ln{10}} + \frac{2 * {10}^{x}ln^{2}(10)*-0}{xln^{2}{10}} - \frac{2*-2 * {10}^{x}ln(10)}{x^{3}ln{10}} - \frac{2({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{x^{2}ln{10}} - \frac{2 * {10}^{x}*0}{x^{2}(10)ln{10}} - \frac{2 * {10}^{x}ln(10)*-0}{x^{2}ln^{2}{10}} + \frac{2*-3 * {10}^{x}}{x^{4}ln{10}} + \frac{2({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{3}ln{10}} + \frac{2 * {10}^{x}*-0}{x^{3}ln^{2}{10}}\\=&{10}^{x}ln^{4}(10)lg(x) + \frac{{10}^{x}ln^{3}(10)}{xln{10}} - \frac{{10}^{x}ln^{2}(10)}{x^{2}ln{10}} + \frac{3 * {10}^{x}ln^{3}(10)}{xln{10}} + \frac{2 * {10}^{x}ln(10)}{x^{3}ln{10}} - \frac{5 * {10}^{x}ln^{2}(10)}{x^{2}ln{10}} + \frac{6 * {10}^{x}ln(10)}{x^{3}ln{10}} - \frac{6 * {10}^{x}}{x^{4}ln{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ ln(x){10}^{x} + \frac{{10}^{x}}{(xln(10))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {10}^{x}ln(x) + \frac{{10}^{x}}{xln(10)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {10}^{x}ln(x) + \frac{{10}^{x}}{xln(10)}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(x) + \frac{{10}^{x}}{(x)} + \frac{-{10}^{x}}{x^{2}ln(10)} + \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{xln(10)} + \frac{{10}^{x}*-0}{xln^{2}(10)(10)}\\=&{10}^{x}ln(10)ln(x) - \frac{{10}^{x}}{x^{2}ln(10)} + \frac{2 * {10}^{x}}{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln(10)ln(x) - \frac{{10}^{x}}{x^{2}ln(10)} + \frac{2 * {10}^{x}}{x}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)ln(x) + \frac{{10}^{x}*0ln(x)}{(10)} + \frac{{10}^{x}ln(10)}{(x)} - \frac{-2 * {10}^{x}}{x^{3}ln(10)} - \frac{({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{2}ln(10)} - \frac{{10}^{x}*-0}{x^{2}ln^{2}(10)(10)} + \frac{2*-{10}^{x}}{x^{2}} + \frac{2({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x}\\=&{10}^{x}ln^{2}(10)ln(x) + \frac{3 * {10}^{x}ln(10)}{x} + \frac{2 * {10}^{x}}{x^{3}ln(10)} - \frac{3 * {10}^{x}}{x^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln^{2}(10)ln(x) + \frac{3 * {10}^{x}ln(10)}{x} + \frac{2 * {10}^{x}}{x^{3}ln(10)} - \frac{3 * {10}^{x}}{x^{2}}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{2}(10)ln(x) + \frac{{10}^{x}*2ln(10)*0ln(x)}{(10)} + \frac{{10}^{x}ln^{2}(10)}{(x)} + \frac{3*-{10}^{x}ln(10)}{x^{2}} + \frac{3({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{x} + \frac{3 * {10}^{x}*0}{x(10)} + \frac{2*-3 * {10}^{x}}{x^{4}ln(10)} + \frac{2({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{3}ln(10)} + \frac{2 * {10}^{x}*-0}{x^{3}ln^{2}(10)(10)} - \frac{3*-2 * {10}^{x}}{x^{3}} - \frac{3({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{2}}\\=&{10}^{x}ln^{3}(10)ln(x) + \frac{4 * {10}^{x}ln^{2}(10)}{x} - \frac{6 * {10}^{x}ln(10)}{x^{2}} - \frac{6 * {10}^{x}}{x^{4}ln(10)} + \frac{8 * {10}^{x}}{x^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( {10}^{x}ln^{3}(10)ln(x) + \frac{4 * {10}^{x}ln^{2}(10)}{x} - \frac{6 * {10}^{x}ln(10)}{x^{2}} - \frac{6 * {10}^{x}}{x^{4}ln(10)} + \frac{8 * {10}^{x}}{x^{3}}\right)}{dx}\\=&({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{3}(10)ln(x) + \frac{{10}^{x}*3ln^{2}(10)*0ln(x)}{(10)} + \frac{{10}^{x}ln^{3}(10)}{(x)} + \frac{4*-{10}^{x}ln^{2}(10)}{x^{2}} + \frac{4({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln^{2}(10)}{x} + \frac{4 * {10}^{x}*2ln(10)*0}{x(10)} - \frac{6*-2 * {10}^{x}ln(10)}{x^{3}} - \frac{6({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))ln(10)}{x^{2}} - \frac{6 * {10}^{x}*0}{x^{2}(10)} - \frac{6*-4 * {10}^{x}}{x^{5}ln(10)} - \frac{6({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{4}ln(10)} - \frac{6 * {10}^{x}*-0}{x^{4}ln^{2}(10)(10)} + \frac{8*-3 * {10}^{x}}{x^{4}} + \frac{8({10}^{x}((1)ln(10) + \frac{(x)(0)}{(10)}))}{x^{3}}\\=&{10}^{x}ln^{4}(10)ln(x) + \frac{5 * {10}^{x}ln^{3}(10)}{x} - \frac{10 * {10}^{x}ln^{2}(10)}{x^{2}} + \frac{20 * {10}^{x}ln(10)}{x^{3}} + \frac{24 * {10}^{x}}{x^{5}ln(10)} - \frac{30 * {10}^{x}}{x^{4}}\\ \end{split}\end{equation} \]