There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ {(cos(x))}^{\frac{1}{ln(1 + {x}^{2})}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {cos(x)}^{\frac{1}{ln(x^{2} + 1)}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {cos(x)}^{\frac{1}{ln(x^{2} + 1)}}\right)}{dx}\\=&({cos(x)}^{\frac{1}{ln(x^{2} + 1)}}((\frac{-(2x + 0)}{ln^{2}(x^{2} + 1)(x^{2} + 1)})ln(cos(x)) + \frac{(\frac{1}{ln(x^{2} + 1)})(-sin(x))}{(cos(x))}))\\=&\frac{-2x{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln(cos(x))}{(x^{2} + 1)ln^{2}(x^{2} + 1)} - \frac{{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}sin(x)}{ln(x^{2} + 1)cos(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln(cos(x))}{(x^{2} + 1)ln^{2}(x^{2} + 1)} - \frac{{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}sin(x)}{ln(x^{2} + 1)cos(x)}\right)}{dx}\\=&\frac{-2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln(cos(x))}{ln^{2}(x^{2} + 1)} - \frac{2{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln(cos(x))}{(x^{2} + 1)ln^{2}(x^{2} + 1)} - \frac{2x({cos(x)}^{\frac{1}{ln(x^{2} + 1)}}((\frac{-(2x + 0)}{ln^{2}(x^{2} + 1)(x^{2} + 1)})ln(cos(x)) + \frac{(\frac{1}{ln(x^{2} + 1)})(-sin(x))}{(cos(x))}))ln(cos(x))}{(x^{2} + 1)ln^{2}(x^{2} + 1)} - \frac{2x{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}*-sin(x)}{(x^{2} + 1)(cos(x))ln^{2}(x^{2} + 1)} - \frac{2x{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln(cos(x))*-2(2x + 0)}{(x^{2} + 1)ln^{3}(x^{2} + 1)(x^{2} + 1)} - \frac{({cos(x)}^{\frac{1}{ln(x^{2} + 1)}}((\frac{-(2x + 0)}{ln^{2}(x^{2} + 1)(x^{2} + 1)})ln(cos(x)) + \frac{(\frac{1}{ln(x^{2} + 1)})(-sin(x))}{(cos(x))}))sin(x)}{ln(x^{2} + 1)cos(x)} - \frac{{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}*-(2x + 0)sin(x)}{ln^{2}(x^{2} + 1)(x^{2} + 1)cos(x)} - \frac{{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}cos(x)}{ln(x^{2} + 1)cos(x)} - \frac{{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}sin(x)sin(x)}{ln(x^{2} + 1)cos^{2}(x)}\\=&\frac{2x{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln(cos(x))sin(x)}{(x^{2} + 1)ln^{3}(x^{2} + 1)cos(x)} - \frac{2{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln(cos(x))}{(x^{2} + 1)ln^{2}(x^{2} + 1)} + \frac{2x{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln(cos(x))sin(x)}{(x^{2} + 1)ln^{3}(x^{2} + 1)cos(x)} + \frac{4x^{2}{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln(cos(x))}{(x^{2} + 1)^{2}ln^{2}(x^{2} + 1)} + \frac{4x{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}sin(x)}{(x^{2} + 1)ln^{2}(x^{2} + 1)cos(x)} + \frac{8x^{2}{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln(cos(x))}{(x^{2} + 1)^{2}ln^{3}(x^{2} + 1)} + \frac{4x^{2}{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}ln^{2}(cos(x))}{(x^{2} + 1)^{2}ln^{4}(x^{2} + 1)} + \frac{{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}sin^{2}(x)}{ln^{2}(x^{2} + 1)cos^{2}(x)} - \frac{{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}sin^{2}(x)}{ln(x^{2} + 1)cos^{2}(x)} - \frac{{cos(x)}^{\frac{1}{ln(x^{2} + 1)}}}{ln(x^{2} + 1)}\\ \end{split}\end{equation} \]

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