For my PDE class, I need to solve the following problem:
Find a function $f(x)$ whose derivative in the sense of distributions is the distribution $x^2+4x+\delta_2$
MY PROOF
$$\langle(F_f)',\phi(x)\rangle=\langle x^2+4x+\delta_2,\phi(x)\rangle \\[3ex] \begin{align*}-\langle F_f,\phi(x)'\rangle&=\left\langle \left(\frac{x^3}{3}+2x^2+H(x-2)\right)',\phi(x)\right\rangle \\[1ex] &=-\left\langle \frac{x^3}{3}+2x^2+H(x-2),\phi'(x)\right\rangle\end{align*} \\[10ex] \Rightarrow \, f(x)=\frac{x^3}{3}+2x^2+H(x-2)$$
I know my final result is correct but I was wondering if my proof was also correct. I feel like I need to detail more how I got from the first line to the second line but I'm not sure how to do it and still be correct. Otherwise, I can always say that $f(x)$ is equal to that and then take the derivative and conclude that $f'(x)$ is indeed equal to $x^2+4x+\delta_2$ in the sense of distributions that but that's much longer to write.
Thanks in advance for your answers