To compute logistic regression and its gradient descent on $m$ examples, we can use for loop to accumulate errors and derivatives and then average them, but it will take a long time to run on a big data set. So vectorization is a good way to get rid of explicit for loop in your code. First, stack the $m$ examples horizontally into vectors of $X$ and $Y$, so the shape of $X$ is $(n_x, m)$, where $n_x$ is the number of features, and the shape of $Y$ is (1, m). Then, compute the $Z$ and $A$, $[z^{(1)}z^{(2)}...z^{(m)}]=Z=w^TX+b$ $=[(w^Tx{(1)}+b)(w^Tx{(2)}+b)...(w^Tx{(m)}+b)]$ $[a{(1)} a{(2)} ... a{(m)}] = A=\sigma (Z)$ , the shape of $Z$ and $A$ should be $(1, m)$. The derivatives of $L$ with respect to $Z$, $w$, and $b$ are $\frac{d\mathcal L}{dz} = A - Y$ $\frac{d\mathcal L}{dw} = \frac{1}{m} X {(\frac{d\mathcal L}{dz})}^T$ $\frac{d\mathcal L}{db} = \frac{1}{m} \sum \frac{d\mathcal L}{dz}$