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

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