There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ ln(24x + {e}^{(-5x)})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(24x + {e}^{(-5x)})\right)}{dx}\\=&\frac{(24 + ({e}^{(-5x)}((-5)ln(e) + \frac{(-5x)(0)}{(e)})))}{(24x + {e}^{(-5x)})}\\=&\frac{-5{e}^{(-5x)}}{(24x + {e}^{(-5x)})} + \frac{24}{(24x + {e}^{(-5x)})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-5{e}^{(-5x)}}{(24x + {e}^{(-5x)})} + \frac{24}{(24x + {e}^{(-5x)})}\right)}{dx}\\=&-5(\frac{-(24 + ({e}^{(-5x)}((-5)ln(e) + \frac{(-5x)(0)}{(e)})))}{(24x + {e}^{(-5x)})^{2}}){e}^{(-5x)} - \frac{5({e}^{(-5x)}((-5)ln(e) + \frac{(-5x)(0)}{(e)}))}{(24x + {e}^{(-5x)})} + 24(\frac{-(24 + ({e}^{(-5x)}((-5)ln(e) + \frac{(-5x)(0)}{(e)})))}{(24x + {e}^{(-5x)})^{2}})\\=&\frac{-25{e}^{(-10x)}}{(24x + {e}^{(-5x)})^{2}} + \frac{240{e}^{(-5x)}}{(24x + {e}^{(-5x)})^{2}} + \frac{25{e}^{(-5x)}}{(24x + {e}^{(-5x)})} - \frac{576}{(24x + {e}^{(-5x)})^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-25{e}^{(-10x)}}{(24x + {e}^{(-5x)})^{2}} + \frac{240{e}^{(-5x)}}{(24x + {e}^{(-5x)})^{2}} + \frac{25{e}^{(-5x)}}{(24x + {e}^{(-5x)})} - \frac{576}{(24x + {e}^{(-5x)})^{2}}\right)}{dx}\\=&-25(\frac{-2(24 + ({e}^{(-5x)}((-5)ln(e) + \frac{(-5x)(0)}{(e)})))}{(24x + {e}^{(-5x)})^{3}}){e}^{(-10x)} - \frac{25({e}^{(-10x)}((-10)ln(e) + \frac{(-10x)(0)}{(e)}))}{(24x + {e}^{(-5x)})^{2}} + 240(\frac{-2(24 + ({e}^{(-5x)}((-5)ln(e) + \frac{(-5x)(0)}{(e)})))}{(24x + {e}^{(-5x)})^{3}}){e}^{(-5x)} + \frac{240({e}^{(-5x)}((-5)ln(e) + \frac{(-5x)(0)}{(e)}))}{(24x + {e}^{(-5x)})^{2}} + 25(\frac{-(24 + ({e}^{(-5x)}((-5)ln(e) + \frac{(-5x)(0)}{(e)})))}{(24x + {e}^{(-5x)})^{2}}){e}^{(-5x)} + \frac{25({e}^{(-5x)}((-5)ln(e) + \frac{(-5x)(0)}{(e)}))}{(24x + {e}^{(-5x)})} - 576(\frac{-2(24 + ({e}^{(-5x)}((-5)ln(e) + \frac{(-5x)(0)}{(e)})))}{(24x + {e}^{(-5x)})^{3}})\\=&\frac{-250{e}^{(-15x)}}{(24x + {e}^{(-5x)})^{3}} + \frac{3600{e}^{(-10x)}}{(24x + {e}^{(-5x)})^{3}} + \frac{375{e}^{(-10x)}}{(24x + {e}^{(-5x)})^{2}} - \frac{17280{e}^{(-5x)}}{(24x + {e}^{(-5x)})^{3}} - \frac{1800{e}^{(-5x)}}{(24x + {e}^{(-5x)})^{2}} - \frac{125{e}^{(-5x)}}{(24x + {e}^{(-5x)})} + \frac{27648}{(24x + {e}^{(-5x)})^{3}}\\ \end{split}\end{equation} \]

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