There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{{e}^{x}}{cos(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{{e}^{x}}{cos(x)}\right)}{dx}\\=&\frac{({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{cos(x)} + \frac{{e}^{x}sin(x)}{cos^{2}(x)}\\=&\frac{{e}^{x}}{cos(x)} + \frac{{e}^{x}sin(x)}{cos^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{e}^{x}}{cos(x)} + \frac{{e}^{x}sin(x)}{cos^{2}(x)}\right)}{dx}\\=&\frac{({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{cos(x)} + \frac{{e}^{x}sin(x)}{cos^{2}(x)} + \frac{({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(x)}{cos^{2}(x)} + \frac{{e}^{x}cos(x)}{cos^{2}(x)} + \frac{{e}^{x}sin(x)*2sin(x)}{cos^{3}(x)}\\=&\frac{2{e}^{x}}{cos(x)} + \frac{2{e}^{x}sin(x)}{cos^{2}(x)} + \frac{2{e}^{x}sin^{2}(x)}{cos^{3}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2{e}^{x}}{cos(x)} + \frac{2{e}^{x}sin(x)}{cos^{2}(x)} + \frac{2{e}^{x}sin^{2}(x)}{cos^{3}(x)}\right)}{dx}\\=&\frac{2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{cos(x)} + \frac{2{e}^{x}sin(x)}{cos^{2}(x)} + \frac{2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(x)}{cos^{2}(x)} + \frac{2{e}^{x}cos(x)}{cos^{2}(x)} + \frac{2{e}^{x}sin(x)*2sin(x)}{cos^{3}(x)} + \frac{2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin^{2}(x)}{cos^{3}(x)} + \frac{2{e}^{x}*2sin(x)cos(x)}{cos^{3}(x)} + \frac{2{e}^{x}sin^{2}(x)*3sin(x)}{cos^{4}(x)}\\=&\frac{4{e}^{x}}{cos(x)} + \frac{8{e}^{x}sin(x)}{cos^{2}(x)} + \frac{6{e}^{x}sin^{2}(x)}{cos^{3}(x)} + \frac{6{e}^{x}sin^{3}(x)}{cos^{4}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{4{e}^{x}}{cos(x)} + \frac{8{e}^{x}sin(x)}{cos^{2}(x)} + \frac{6{e}^{x}sin^{2}(x)}{cos^{3}(x)} + \frac{6{e}^{x}sin^{3}(x)}{cos^{4}(x)}\right)}{dx}\\=&\frac{4({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{cos(x)} + \frac{4{e}^{x}sin(x)}{cos^{2}(x)} + \frac{8({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(x)}{cos^{2}(x)} + \frac{8{e}^{x}cos(x)}{cos^{2}(x)} + \frac{8{e}^{x}sin(x)*2sin(x)}{cos^{3}(x)} + \frac{6({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin^{2}(x)}{cos^{3}(x)} + \frac{6{e}^{x}*2sin(x)cos(x)}{cos^{3}(x)} + \frac{6{e}^{x}sin^{2}(x)*3sin(x)}{cos^{4}(x)} + \frac{6({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin^{3}(x)}{cos^{4}(x)} + \frac{6{e}^{x}*3sin^{2}(x)cos(x)}{cos^{4}(x)} + \frac{6{e}^{x}sin^{3}(x)*4sin(x)}{cos^{5}(x)}\\=&\frac{12{e}^{x}}{cos(x)} + \frac{24{e}^{x}sin(x)}{cos^{2}(x)} + \frac{40{e}^{x}sin^{2}(x)}{cos^{3}(x)} + \frac{24{e}^{x}sin^{3}(x)}{cos^{4}(x)} + \frac{24{e}^{x}sin^{4}(x)}{cos^{5}(x)}\\ \end{split}\end{equation} \]

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