Two improper integrals with a parameter converge to the same function

Question

Suppose $\int _{a}^{+\infty}f(x,y)\ \mathrm{d}y$ and $\int _{a}^{+\infty}g(x,y)\ \mathrm{d} y$ both converge to $F(x)$ on X.That is $$F(x)=\int _{a}^{+\infty}f(x,y)\ \mathrm{d}y=\int _{a}^{+\infty}g(x,y)\ \mathrm{d}y,x\in X$$ If $F(x)$ is differentiable,and $\int _{a}^{+\infty}g'_x(x,y)\ \mathrm{d}y$ converges uniformly to $F'(x)$ for $x\in X$. Then could we calculate the derivative of $\int _{a}^{+\infty}f(x,y)\ \mathrm{d}y$ by differentiation under the integral sign?i.e,Could we conclude that $$\frac{\mathrm{d}}{\mathrm{d}x}\int _{a}^{+\infty}f(x,y)\ \mathrm{d}y=\int_a^{+\infty}\frac{\mathrm{\partial}}{\mathrm{\partial}x}f(x,y)\ \mathrm{d}y$$