Is my Gradient and Hessian of the following correct? \begin{align} f &= \sum_i \log \left(1 + \exp\left\{ -t_i \left(w^T x_i\right)\right\} \right) + \mu \|w \|_2^2 \ , \end{align} where $t_i \in \mathbb{R}$, $w, x_i \in \mathbb{R}^n$, and $\mu \in \mathbb{R}$.
I want to find the gradient and Hessian w.r.t. $w$, that is $\frac{\partial f}{\partial w}$ and $\frac{\partial^2 f}{\partial w^2}$.
Partial attempt
Gradient
\begin{align} \frac{\partial f}{\partial w} &= \sum_i \left( \frac{-t_i x_i \exp\left\{ -t_i \left(w^T x_i\right)\right\}}{1 + \exp\left\{ -t_i \left(w^T x_i\right)\right\} } \right) + 2\mu w \ \ \\ &= \sum_i \left( \frac{-t_i x_i }{1 + \exp\left\{ +t_i \left(w^T x_i\right)\right\} } \right) + 2\mu w \ . \end{align}
is this Gradient correct?
Hessian
\begin{align} \frac{\partial^2 f}{\partial w^2} &= \frac{\partial}{ \partial w} \left[ \sum_i \left( \frac{-t_i x_i }{1 + \exp\left\{ +t_i \left(w^T x_i\right)\right\} } \right) + 2\mu w \right] \\ &= \sum_i \left( \frac{t_i^2 x_i x_i \ \exp\left\{ +t_i \left(w^T x_i\right)\right\}}{\left(1 + \exp\left\{ +t_i \left(w^T x_i\right)\right\}\right)^2 } \right) + 2\mu I \ . \end{align}
I Think my Hessian is for sure incorrect, isn't it? because I am getting in the numerator of the first part as $x_i x_i$... how would two vectors just multiply :( ...