Evaluate a derivative of the function $f(x,y)=(\int_0^{x+y} \varphi, \int_{0}^{xy} \varphi)$

Question

The problem is the following. Evaluate the derivative of the function $f(x,y)=(\int_0^{x+y} \varphi, \int_{0}^{xy} \varphi)$ in the point $(a,b)$, where $\varphi$ is integrable and continuous.

The basic approach that I have to use the definition of the derivative of $f$, it is calculate the matrix

$$A=\begin{pmatrix} \frac{\partial f_1}{\partial x} && \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} && \frac{\partial f_2}{\partial y}\end{pmatrix}$$

where $f_1$ and $f_2$ are the components of the function $f$. Now, in this point I have problems in the calculation of the partial derivatives of the components of $f$. For instance, I want evaluate $$\frac{\partial}{\partial x}\int_{0}^{x+y}\varphi$$

I think that I should use the Leibnitz rule (in the only form that I know) which state that under certain hypothesis we have that $$\frac{d}{dx}\int_{a(x)}^{b(x)}f(x,t) dt=f(x,b(x))b^{\prime}(x)-f(x,a(x))a^{\prime}(x)+\int_{a(x)}^{b(x)}\frac{\partial}{\partial x}f(x,t)dt$$

Now in order to apply this rule I have a problem in the limits that I currently have, which are in terms of two variables. So the questions are

1. How to evaluate the specific integral $$\frac{\partial}{\partial x}\int_{0}^{x+y}\varphi$$

2. How to evaluate a more general integral of the kind

$$\frac{\partial}{\partial x}\int_{a(x,y)}^{b(x,y)}\varphi$$ or $$\frac{\partial}{\partial y}\int_{a(x,y)}^{b(x,y)}\varphi$$

3. There is a general form of the Leibnitz rule as I expect?

4. In the case that the current approach that I use were not correct, what should be the right approach?

I appreciate any suggestion of the community

Current situation of the problem

By Pedro comment, this particular problem is more easy by the easy remark that $$F(x)=\int_0^x \varphi(t)dt$$ and identify each component function with $F(x+y)$ and $F(xy)$ and use the usual Leibnitz rule to compute for instance the matrix $$A= \begin{pmatrix} \varphi(x+y) && \varphi(x+y) \\ y\varphi(xy) && x\varphi(xy)\end{pmatrix}$$

Which actually help me a lot, but now I can not identify how to answer question 2, it is when we can not use the remark of Pedro, for instance, what if I want to evaluate $$\frac{\partial}{\partial x}\int_{e^x}^{\sqrt{y}}\varphi$$ or more generally $$\frac{\partial}{\partial x}\int_{a(x,y)}^{b(x,y)}\varphi$$

Answer 1

$$\frac{\partial F}{\partial x}(x_0,y_0)=\frac{dF(\cdot,y_0)}{dx}(x_0),$$ so you don't need anything more than the Leibniz rule you already know.

For instance, for $f_2(x,y)=\int_0^{xy}\varphi,$ $$\frac{\partial f_2}{\partial x}(x_0,y_0)=\left(\frac d{dx}\int_0^{xy_0}\varphi(t)dt\right)(x_0)=\left(\varphi(xy_0)\frac d{dx}(xy_0)\right)(x_0)=\varphi(x_0y_0)y_0,$$ by the following particular case of your Leibniz rule: $$\frac{d}{dx}\int_0^{b(x)}f(t) dt=f(b(x))b^{\prime}(x).$$

More generally, similarly, $$\left(\frac{\partial}{\partial x}\int_{a(x,y)}^{b(x,y)}\varphi\right)(x_0,y_0)=\left(\frac d{dx}\int_{a(x,y_0)}^{b(x,y_0)}\varphi\right)(x_0)$$

$$=\left(\varphi(b(x,y_0))\frac{db(x,y_0)}{dx}-\varphi(a(x,y_0))\frac{da(x,y_0)}{dx}\right)(x_0)$$

$$=\varphi(b(x_0,y_0))\frac{\partial b}{\partial x}(x_0,y_0)-\varphi(a(x_0,y_0))\frac{\partial a}{\partial x}(x_0,y_0).$$

Answer 2

It is a bit simpler than you think. Notice that if $F(x) = \int_0^x \varphi(t)dt$ then $F'(x) = \varphi(x)$. Then, by the chain rule, $F(x+y)$ has derivative $\varphi(x+y)$ with respect to $x$, since $y$ is just constant. Similarly, $F(xy)$ has derivative $F'(xy)y = \varphi(xy)y$ with respect to $x$ since $y$ is just a constant. By symmetry, you can compute the partial derivatives with respect to $y$.