There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ ln({(\frac{(1 + sin(x))}{cos(x)})}^{\frac{1}{2}}) - \frac{tan(x)}{(2{(sin(x))}^{2})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln((\frac{sin(x)}{cos(x)} + \frac{1}{cos(x)})^{\frac{1}{2}}) - \frac{\frac{1}{2}tan(x)}{sin^{2}(x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln((\frac{sin(x)}{cos(x)} + \frac{1}{cos(x)})^{\frac{1}{2}}) - \frac{\frac{1}{2}tan(x)}{sin^{2}(x)}\right)}{dx}\\=&\frac{(\frac{\frac{1}{2}(\frac{cos(x)}{cos(x)} + \frac{sin(x)sin(x)}{cos^{2}(x)} + \frac{sin(x)}{cos^{2}(x)})}{(\frac{sin(x)}{cos(x)} + \frac{1}{cos(x)})^{\frac{1}{2}}})}{((\frac{sin(x)}{cos(x)} + \frac{1}{cos(x)})^{\frac{1}{2}})} - \frac{\frac{1}{2}*-2cos(x)tan(x)}{sin^{3}(x)} - \frac{\frac{1}{2}sec^{2}(x)(1)}{sin^{2}(x)}\\=&\frac{sin^{2}(x)}{2(\frac{sin(x)}{cos(x)} + \frac{1}{cos(x)})cos^{2}(x)} + \frac{sin(x)}{2(\frac{sin(x)}{cos(x)} + \frac{1}{cos(x)})cos^{2}(x)} + \frac{1}{2(\frac{sin(x)}{cos(x)} + \frac{1}{cos(x)})} + \frac{cos(x)tan(x)}{sin^{3}(x)} - \frac{sec^{2}(x)}{2sin^{2}(x)}\\ \end{split}\end{equation} \]

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