= Solution
The required anomaly cancellation depends on the exact <Eisenstein series of weight two> law
$$
E_2(\gamma\tau)=j^2E_2(\tau)+\frac{12c}{2\pi i}j,\qquad j=c\tau+d.
$$
Here is a way to establish it without assuming that $E_2$ is a <modular form>. The lattice definition of the <completed nonholomorphic Eisenstein series> in Question 6 is <modular invariant>. Its <Poisson summation> calculation, proved there, gives the finite Laurent coefficient at $s=1$ as
$$
C(\tau)=\frac{\gamma_E-\log(4\pi)}2-\frac12\log y+\frac{\pi y}{6}-2\sum_{n\geq1}\log|1-q^n|,\qquad y=\Im\tau.
$$
The residue is constant, so $C$ is also invariant. This calculation uses only convergent Gaussian integrals and the analytic infinite product, not modularity of that product. Using the <Wirtinger derivative> $\partial_\tau$, with $\partial_\tau y=1/(2i)$, differentiation of the locally uniformly convergent series gives
$$
\partial_\tau C=-\frac{\pi i}{12}\left(E_2(\tau)-\frac3{\pi y}\right).
$$
Differentiate $C(\gamma\tau)=C(\tau)$ to see that $E_2^*=E_2-3/(\pi y)$ transforms with weight two. Substituting $\Im(\gamma\tau)=y/|j|^2$ yields
$$
E_2(\gamma\tau)-j^2E_2(\tau)=\frac3{\pi y}(|j|^2-j^2)
=\frac{12c}{2\pi i}j,
$$
since $\overline j-j=-2icy$. This is the <weight-two transformation from an invariant Eisenstein limit>.
Combine this law with the <derivative transformation of a weak modular form> proved in part (i). The anomalous terms in $Df-(k/12)E_2f$ cancel exactly, leaving the weight-$k+2$ transformation law. Both functions are holomorphic on the <complex upper half-plane>, and their <Fourier expansions> contain no negative powers. Hence the <Serre derivative> satisfies
$$
\boxed{D_kf:=Df-\frac{k}{12}E_2f\in M_{k+2}.}
$$
Its constant coefficient is $-ka_0(f)/12$; it need not be a <cusp form> when $f$ is not cuspidal.
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