Require
$$
(\partial_\mu+A'_\mu)(g\phi)=g(\partial_\mu+A_\mu)\phi
$$
for every $\phi$. Expanding and cancelling $g\partial_\mu\phi$ gives
$$
(\partial_\mu g)+A'_\mu g=gA_\mu.
$$
Right multiplication by $g^{-1}$ yields the <gauge-field transformation law>
$$
A'_\mu=gA_\mu g^{-1}-(\partial_\mu g)g^{-1}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 4 / a / Solution

For $D_\mu=\partial_\mu+A_\mu$ and a matter field transforming as $\phi\mapsto g\phi$, covariance of $D_\mu\phi$ requires
$$
A_\mu\mapsto gA_\mu g^{-1}-(\partial_\mu g)g^{-1}.
$$


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 Incoming links: Gauge-field transformation law