= Solution
Let $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb Z)$ and $j=c\tau+d$. Differentiating the transformation law of a weight-$k$ <modular form> gives the <derivative transformation of a weak modular form>
$$
(Df)(\gamma\tau)=j^{k+2}Df(\tau)+\frac{kc}{2\pi i}j^{k+1}f(\tau).
$$
The corresponding anomaly for $Dg$ has coefficient $\ell c/(2\pi i)$. Consequently these anomalies cancel in $\ell gDf-kfDg$, which transforms with weight $k+\ell+2$. The <derivatives> are holomorphic on the <complex upper half-plane>. In a <Fourier expansion>, $D(\sum_{n\geq0}a_nq^n)=\sum_{n\geq1}na_nq^n$, so each term in the combination has zero constant coefficient and no negative powers. There is just one <modular cusp> up to the full <modular group>. Therefore
$$
\boxed{\ell gDf-kfDg\in S_{k+\ell+2}.}
$$
This is, up to the chosen sign, the <first Rankin-Cohen bracket>.
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