Solution
= Solution
Under the regularity and integrability conditions for the <Leibniz integral rule>,
$$
\boxed{\frac{d}{d\theta}\int_{a(\theta)}^{b(\theta)}f(x,\theta)\,dx
=f(b(\theta),\theta)b'(\theta)-f(a(\theta),\theta)a'(\theta)
+\int_{a(\theta)}^{b(\theta)}\frac{\partial f}{\partial\theta}(x,\theta)\,dx}.
$$
The first two terms account for the moving endpoints; the last accounts for the integrand's own parameter dependence.