I'm trying to solve the following problem:
Let $2\leq m$ and let $M_m$ be the vector space of modular forms of weight $m$ for $SL_2(Z)$. Show that for $f\in M_k$ and $ g\in M_l$ the following holds:
i) $lf'g-kfg'\in M_{l+k+2}$
ii) $l(l+1)f''g-2(k+1)(l+1)f'g'+k(k+1)fg''\in M_{l+k+4}$
Edit: Sorry I didn't write more from the start.
I'm somewhat new to modular forms, I'm mostly using Freitags "Complex Analysis" to teach myself. He lists two exercises that are somewhat similar to what I'm trying to show:
Let f be a modular function, show that its derivative is a modular form of weight 2.
The solution to this one is somewhat easy, I got to it by just differtiating with the chain rule. Trying the same with forms of higher weight, I found that they aren't modular forms.
The other exercise is:
Let $f$ and $g$ be modular forms of weight $k$, show that $f'g-fg'$ is a modular form of weight $2k+2$.
The solution to that exercise is using an identity ($f'g-fg'=-f^2(\frac{g}{f})'$) I don't know how to prove, though the answer is easily derived from that if I accept it to be true.
Back to my original problem I've tried to adapt the identity used in the exercise to derive the result but I came to nothing solid. I've also tried to use the fact that modular forms obey $f(z+1)=f(z)$ and $f(-\frac{1}{z})=z^k f(z)$ to just use the definition of a modular form. The first part is immediate from the definition of $lf'g-kfg'$, so I tried to verify the second equality but got stuck in the algebra.