Logistic Regression Manual Update
The update rules are ;
$a := a - \alpha.\frac{\partial L}{\partial a}$
$b := b - \alpha.\frac{\partial L}{\partial b}$
Here L which is the Loss [Squared Error in this case ] is given by;
$L  = \frac{1}{2}.(\hat{y} - \frac{1.0}{(1.0 + exp^{-ax + b})})^{2}$
So ;
$\frac{\partial L}{\partial a} = \frac{1}{2}.2(\hat{y} - \frac{1.0}{(1.0 + exp^{-ax + b})}).\frac{-1.0}{(1.0 + exp^{-ax + b})^{2}}.(-x).e^{-ax+b}  $
$\frac{\partial L}{\partial b} = \frac{1}{2}.2(\hat{y} - \frac{1.0}{(1.0 + exp^{-ax + b})}).\frac{-1.0}{(1.0 + exp^{-ax + b})^{2}}.e^{-ax+b} $
So now guess an intial value for a and b.
You can plug in the an x and y pair for and update the weights simultaneously. The thing to be careful is that when updating for a and b in an iteration use the previous a and b values. That is if you have updated your a prior to b dont use the updated value of a to update b.