Leibniz integral rule for differentiation under the integral sign

In calculus the Leibniz integral rule for differentiation under the integral sign is based on the following theorem. 

Theorem for differentiation under the integral sign:

Suppose $a \le b$ and $c \le d$ are four real constants. 

Suppose also that $f(x,y) : P \to \mathbb{R}$ is a function which is continuous and has a continuous partial derivative $f_y(x,y)$ in the rectangle 

$$P = \{(x,y) | a \le x \le b, c \le y \le d\} \tag{1}$$

Suppose also that the functions 

$$\alpha(y), \beta(y) : [c,d] \to [a,b]$$ 

are differentiable in the interval $[c,d]$. Then the integral 

$$\Phi(y) = \int\limits_{\alpha(y)}^{\beta(y)} f(x,y) \ dx$$

(which is a function of $y \in [c,d]$) is a differentiable function of $y$ in the interval $[c,d]$ and 

$$\Phi'(y) = f(\beta(y),y)\cdot \beta'(y) - f(\alpha(y),y)\cdot \alpha'(y) + \int\limits_{\alpha(y)}^{\beta(y)} f'_{y}(x,y) \ dx \tag{2}$$

Note:

In the special case when $\alpha(y) = \alpha$ and $\beta(y) = \beta$ are constant functions equality $(2)$ takes the form

$$\Phi'(y) = \int\limits_{\alpha}^{\beta} f'_{y}(x,y) \ dx \tag{3}$$

because $\alpha '(y) = \beta '(y) = 0$. In some books this special case is also known as theorem for differentiation under the integral sign.