I am not sure how to solve this partial derivative in respect to $\theta$. The portion that is confusing me is the summation portion. I assume that because the summation does not have $\theta$ we can just "pull it out" and it is not in respect to $\theta$.
Here we have $\langle\theta,x^i\rangle$ representing the dot product. I have that portion figured out.
$$\dfrac{\partial\sum_{i=0}^n log(1 + exp(y^i\langle\theta,x^i\rangle))}{\partial\theta}$$
From my understanding
$$\dfrac{\partial\sum_{i=0}^n log(1 + exp(y^i\langle\theta,x^i\rangle))}{\partial\theta}$$
would be treated the same as
$$\dfrac{\sum_{i=0}^n \partial log(1 + exp(y^i\langle\theta,x^i\rangle))}{\partial\theta}$$
where I can start solving and ignoring the summation ($\sum_{i=0}^n$) portion
Let's differentiate with respect to $\theta_j$.
\begin{align}\dfrac{\sum_{i=0}^n \partial \log(1 + \exp(y^i\langle\theta,x^i\rangle))}{\partial\theta_j} &= \sum_{i=1}^n\frac{y^ix^i_j}{1+\exp(y^i\langle \theta, x^i\rangle)}\\ \end{align}
We have found the $j$-th component of the derivative.