Derivative of CARA utility
All you need for this particular question is the following. Let $\mathbf{X}$ be a $T \times K$ matrix, $\mathbf{w}$ a K-dimensional vector and $\mathbf{y}$ a T-dimensional vector, then

$$
\begin{eqnarray*}
\frac{\partial \mathbf{w}^{\prime}\mathbf{X}^{\prime}\mathbf{y}}{\partial \mathbf{w}}&amp;=&amp;
\mathbf{X}^{\prime}\mathbf{y}\\
\frac{\partial \mathbf{w}^{\prime}\mathbf{X}^{\prime}\mathbf{X}\mathbf{w}}{\partial\mathbf{w}}&amp;=&amp;2\mathbf{X}^{\prime}\mathbf{X}\mathbf{w}
\end{eqnarray*}
$$
Back to your original post, we will first rewrite it in order to apply the rules provided. The first element in the expression we want to differentiate is the same as $\phi'(\mu-R_f\mathbf{1})$, hence
$$
\frac{\partial \phi'(\mu-R_f\mathbf{1})-1/2\alpha\phi'\Sigma\phi}{\partial \phi}=
(\mu-R_f\mathbf{1})-\alpha\Sigma\phi
$$
and equating to 0 you get the desired result.