Differentiation under the integral sign

Statement

Let $f \colon [a, b] \times [t_0, t_1] \to \mathbb{R}$ be continuous, and suppose that its partial derivative $\partial f / \partial t$ exists and is continuous on $[a, b] \times [t_0, t_1]$. Then $F(t) = \int_a^b f(x, t) \, dx$ is continuously differentiable on $[t_0, t_1]$, and $$F'(t) = \int_a^b \frac{\partial f}{\partial t}(x, t) \, dx.$$ If, in addition, $\alpha, \beta \colon [t_0, t_1] \to [a, b]$ are continuously differentiable, then $$\frac{d}{dt} \int_{\alpha(t)}^{\beta(t)} f(x, t) \, dx = \int_{\alpha(t)}^{\beta(t)} \frac{\partial f}{\partial t}(x, t) \, dx + f(\beta(t), t) \, \beta'(t) - f(\alpha(t), t) \, \alpha'(t).$$