Error in differentiation/integration problem (Edited to directly present the problem only, for any future readers. The original question can be read in the revision history.)
Is the following correct?
$$\frac{d}{dy}\left(\int y\,dx\right) = x$$
If not what is the correct solution?

Let's use a bit more precise notation and define $f(x)=x^3+x^2$.  Then, with $y=f(x)$, we can write 
$$\begin{align}
z(x)&=\frac{d}{dy}\int f(x)\,dx\\\\
&=\left . \left(\frac{d}{dy}\int y\,\frac{df^{-1}(y)}{dy}\,dy\right)\right|_{y=f(x)}\\\\
&=\left.   \left(y\frac{df^{-1}(y)}{dy}\right)\right|_{y=f(x)}
\end{align}$$
