There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ sin(e^{x} + lg(y))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(e^{x} + lg(y))\right)}{dx}\\=&cos(e^{x} + lg(y))(e^{x} + \frac{0}{ln{10}(y)})\\=&e^{x}cos(e^{x} + lg(y))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{x}cos(e^{x} + lg(y))\right)}{dx}\\=&e^{x}cos(e^{x} + lg(y)) + e^{x}*-sin(e^{x} + lg(y))(e^{x} + \frac{0}{ln{10}(y)})\\=&e^{x}cos(e^{x} + lg(y)) - e^{{x}*{2}}sin(e^{x} + lg(y))\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{x}cos(e^{x} + lg(y)) - e^{{x}*{2}}sin(e^{x} + lg(y))\right)}{dx}\\=&e^{x}cos(e^{x} + lg(y)) + e^{x}*-sin(e^{x} + lg(y))(e^{x} + \frac{0}{ln{10}(y)}) - 2e^{x}e^{x}sin(e^{x} + lg(y)) - e^{{x}*{2}}cos(e^{x} + lg(y))(e^{x} + \frac{0}{ln{10}(y)})\\=&e^{x}cos(e^{x} + lg(y)) - 3e^{{x}*{2}}sin(e^{x} + lg(y)) - e^{{x}*{3}}cos(e^{x} + lg(y))\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{x}cos(e^{x} + lg(y)) - 3e^{{x}*{2}}sin(e^{x} + lg(y)) - e^{{x}*{3}}cos(e^{x} + lg(y))\right)}{dx}\\=&e^{x}cos(e^{x} + lg(y)) + e^{x}*-sin(e^{x} + lg(y))(e^{x} + \frac{0}{ln{10}(y)}) - 3*2e^{x}e^{x}sin(e^{x} + lg(y)) - 3e^{{x}*{2}}cos(e^{x} + lg(y))(e^{x} + \frac{0}{ln{10}(y)}) - 3e^{{x}*{2}}e^{x}cos(e^{x} + lg(y)) - e^{{x}*{3}}*-sin(e^{x} + lg(y))(e^{x} + \frac{0}{ln{10}(y)})\\=&e^{x}cos(e^{x} + lg(y)) - 7e^{{x}*{2}}sin(e^{x} + lg(y)) - 6e^{{x}*{3}}cos(e^{x} + lg(y)) + e^{{x}*{4}}sin(e^{x} + lg(y))\\ \end{split}\end{equation} \]

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