Differentiation under the integral sign using Fubini.

Question

Let's consider a function $$ f:\mathbb{R}^2\to\mathbb{R} $$ And suppose it is smooth, then for a given $M>0$ and $y\geq 0$, according to Fubini's theorem, we have that $$ \int_{-M}^M\partial_yf(x,y)dx=\frac{d}{dy}\int_0^y\int_{-M}^M\partial_yf(x,y)dxdy=\frac{d}{dy}\int_{-M}^M\int_0^y\partial_yf(x,y)dydx=\frac{d}{dy}\int_{-M}^Mf(x,y)dx $$ Then when $M\to+\infty$ if the limit exists we would have that $$ \int_\mathbb{R}\partial_yf(x,y)dx=\frac{d}{dy}\int_\mathbb{R}f(x,y)dx $$ I would like to know if the reasoning above is correct, any hint would be appreciated.