\begin{xmpl}
For fixed $x$ consider:
    
    \begin{eqnarray*}
    f(p) &=& \ln(p^{x} (1-p)^{n-x})\\
         &=& \ln p^{x} + \ln(1-p)^{n-x}\\%ln of a product is the sum of ln
         &=& x \ln p + (n-x) \ln (1-p)\\
    \end{eqnarray*}
    
    \begin{eqnarray*}
  f'(p)&=& x \frac{1}{p} + \frac{n-x}{1-p}(-1)\\
         &=& \frac{x}{p} - \frac{n-x}{1-p}\\
    \end{eqnarray*}

\end{xmpl}
\begin{xmpl}
    $f(b) = (y-bx)^2$ ($y,x$ fixed)\\
    
    \begin{eqnarray*}
    f'(b)&=& 2 (y-bx) (-x)\\
         &=& -2x (y-bx)\\
         &=&(-2xy) + (2x^2)b
    \end{eqnarray*}
\end{xmpl}