To compute logistic regression and its gradient descent on $m$ examples, using a for loop to accumulate errors and derivatives takes significant time on large datasets. Vectorization efficiently eliminates the need for explicit for loops. First, stack the $m$ examples horizontally into matrices $X$ 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 $Z$ and $A$:

$Z = [z^{(1)} \dots z^{(m)}] = w^T X + b = [(w^T x^{(1)} + b) \dots (w^T x^{(m)} + b)]$

$A = [a^{(1)} \dots a^{(m)}] = \sigma(Z)$

The shape of $Z$ and $A$ is $(1, m)$. The derivatives of the loss function $\mathcal{L}$ with respect to $Z$, $w$, and $b$ are:

$\frac{d\mathcal{L}}{dZ} = A - Y$

$\frac{d\mathcal{L}}{dw} = \frac{1}{m} X \left(\frac{d\mathcal{L}}{dZ}\right)^T$

$\frac{d\mathcal{L}}{db} = \frac{1}{m} \sum \frac{d\mathcal{L}}{dZ}$