This is not a Obsidian/MathJax problem, it’s just how the LaTeX aligned environment works.
I’m not sure what you’re trying to do here, but generally, putting text among equations using \text{} within an aligned environment is not a good practice. Moreover, since every equation in your sequence is somewhat independent of the previous ones (i.e: they are not a system of equations), I don’t see the need for an aligned environment.
I would simply type the text intermissions as regular markdown (you can use $ in it to wrap variable names), and the equations in separate $$ displayed blocks. I.e:
$$f_i(\vec{v})=\sum_{s=1}^{J} A_{is} v_s$$
Therefore:
$$\frac{\partial f_{i}}{\partial A_{i}} = \left(
\frac{\partial f_{i}}{\partial A_{i 1}}, \frac{\partial f_{i}}{\partial A_{i 2}},
\dots, \frac{\partial f_{i}}{\partial A_{i j}}
\right)$$
Since $f_i$ is the sum of the terms of the row $i$ of $A$ and the components of $v$
$$\frac{\partial f_{i}}{\partial A_{i}} = \left(v_1, v_2, v_3, \dots ,v_j\right) = v^T$$
However if we differentiate the $i$-th row of the output vector
with respect to a $j$ row of matrix a different to $i$
$$\frac{\partial f_{i}}{\partial A_{j}} = \left(
\frac{\partial f_{i}}{\partial A_{j 1}}, \frac{\partial f_{i}}{\partial A_{j 2}},
\dots, \frac{\partial f_{i}}{\partial A_{j J}}
\right)$$
Since for $f_{i}$ we only multiply by the elements of row $i$,
differentiating with respect to the row $j$ does not affect the outcome
$$\frac{\partial f_{i}}{\partial A_{j}} = \vec0_{J}^T$$
Note: I also removed \pmatrix and used \left(, \right) instead, which ensures a more consistent size of the fractions inside.
This produces the following output. Note that every $$ block is by default horizontally centered:
The centered display of equations is standard practice in math and science books and papers, so it looks ok to me.
But, if you insist on having all left-aligned (which, in my opinion, looks worse), you can use this css snippet:
.math-block .MathJax {
text-align: left !important;
display: inline !important;
}
And then, the result would be:
I think that at the very least a small left indentation for the displayed formulae would be necessary to make them easier on the eye. For example:
.math-block .MathJax {
text-align: left !important;
display: inline !important;
margin-left: 1em !important;
}
Result: