There are 1 questions in this calculation: for each question, the 1 derivative of f is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ pln(1 + fu) + (1 - p)ln(1 - fd)\ with\ respect\ to\ f：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = pln(uf + 1) + ln(-df + 1) - pln(-df + 1)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( pln(uf + 1) + ln(-df + 1) - pln(-df + 1)\right)}{df}\\=&\frac{p(u + 0)}{(uf + 1)} + \frac{(-d + 0)}{(-df + 1)} - \frac{p(-d + 0)}{(-df + 1)}\\=&\frac{pu}{(uf + 1)} - \frac{d}{(-df + 1)} + \frac{pd}{(-df + 1)}\\ \end{split}\end{equation} \]

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