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

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