There are 5 questions in this calculation: for each question, the 4 derivative of _ is calculated.
Note that variables are case sensitive.\[ \begin{equation}\begin{split}[1/5]Find\ the\ 4th\ derivative\ of\ function\ {x}^{x}{(ln({x}^{2}))}^{2}\ with\ respect\ to\ _:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {x}^{x}ln^{2}(x^{2})\\&\color{blue}{The\ first\ derivative\ function:}\\&\frac{d\left( {x}^{x}ln^{2}(x^{2})\right)}{d_}\\=&({x}^{x}((0)ln(x) + \frac{(x)(0)}{(x)}))ln^{2}(x^{2}) + \frac{{x}^{x}*2ln(x^{2})*0}{(x^{2})}\\=&0\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\ \end{split}\end{equation} \]\[ \begin{equation}\begin{split}[2/5]Find\ the\ 4th\ derivative\ of\ function\ {sin(x)}^{({e}^{x}cos(x))}\ with\ respect\ to\ _:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function:}\\&\frac{d\left( {sin(x)}^{({e}^{x}cos(x))}\right)}{d_}\\=&({sin(x)}^{({e}^{x}cos(x))}((({e}^{x}((0)ln(e) + \frac{(x)(0)}{(e)}))cos(x) + {e}^{x}*-sin(x)*0)ln(sin(x)) + \frac{({e}^{x}cos(x))(cos(x)*0)}{(sin(x))}))\\=&0\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\ \end{split}\end{equation} \]\[ \begin{equation}\begin{split}[3/5]Find\ the\ 4th\ derivative\ of\ function\ {cos({2}^{{x}^{2}})}^{3}\ with\ respect\ to\ _:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = cos^{3}({2}^{x^{2}})\\&\color{blue}{The\ first\ derivative\ function:}\\&\frac{d\left( cos^{3}({2}^{x^{2}})\right)}{d_}\\=&-3cos^{2}({2}^{x^{2}})sin({2}^{x^{2}})({2}^{x^{2}}((0)ln(2) + \frac{(x^{2})(0)}{(2)}))\\=&\frac{0}{2}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{0}{2}\right)}{d_}\\=&0\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\ \end{split}\end{equation} \]\[ \begin{equation}\begin{split}[4/5]Find\ the\ 4th\ derivative\ of\ function\ arctan({x}^{2} + {x}^{1} + 1)\ with\ respect\ to\ _:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = arctan(x^{2} + x + 1)\\&\color{blue}{The\ first\ derivative\ function:}\\&\frac{d\left( arctan(x^{2} + x + 1)\right)}{d_}\\=&(\frac{(0 + 0 + 0)}{(1 + (x^{2} + x + 1)^{2})})\\=&0\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\ \end{split}\end{equation} \]\[ \begin{equation}\begin{split}[5/5]Find\ the\ 4th\ derivative\ of\ function\ ln({1}^{({2}^{({3}^{({4}^{x} + 1)} + 1)} + 1)} + 1)\ with\ respect\ to\ _:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function:}\\&\frac{d\left( ln({1}^{({2}^{({3}^{({4}^{x} + 1)} + 1)} + 1)} + 1)\right)}{d_}\\=&\frac{(({1}^{({2}^{({3}^{({4}^{x} + 1)} + 1)} + 1)}((({2}^{({3}^{({4}^{x} + 1)} + 1)}((({3}^{({4}^{x} + 1)}((({4}^{x}((0)ln(4) + \frac{(x)(0)}{(4)})) + 0)ln(3) + \frac{({4}^{x} + 1)(0)}{(3)})) + 0)ln(2) + \frac{({3}^{({4}^{x} + 1)} + 1)(0)}{(2)})) + 0)ln(1) + \frac{({2}^{({3}^{({4}^{x} + 1)} + 1)} + 1)(0)}{(1)})) + 0)}{({1}^{({2}^{({3}^{({4}^{x} + 1)} + 1)} + 1)} + 1)}\\=&0\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{d_}\\=&0\\ \end{split}\end{equation} \]Your problem has not been solved here? Please go to the Hot Problems section!