Error in differentiation/integration problem

Question

(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?

Answer 1

Let's use a bit more precise notation and define $f(x)=x^3+x^2$. Then, with $y=f(x)$, we can write

$$\begin{align} z(x)&=\frac{d}{dy}\int f(x)\,dx\\\\ &=\left . \left(\frac{d}{dy}\int y\,\frac{df^{-1}(y)}{dy}\,dy\right)\right|_{y=f(x)}\\\\ &=\left. \left(y\frac{df^{-1}(y)}{dy}\right)\right|_{y=f(x)} \end{align}$$