derivative under integral intuition

Question

let $f(x,y)$ continuous in $[a,b] \times [c,d]$ and $\frac{\partial f}{\partial y} $ continuous in $[a,b] \times [c,d]$, define $$F(y) = \int_a^b f(x,y)dx$$ then $$F'(y) = \int_a^b \frac{\partial f}{\partial y}(x,y) dx$$

Can you explain the intuition behind this rule, in terms of geometry for example, or how did Leibniz came up with this rule?

Answer 1

To understand some formulas from calculus intuitively, we could try to interpret the notions in discrete ways.

For $y\in [c,d]$, the values

$$F(y) = \int_a^b f(x,y)dx$$

denote the areas of the slices (integrals along the x axis) specified by $y$:

To measure the difference between each two adjacent areas (discrete analog of $F'(y)$), one tried to draw all the little changes in height between the slices.

In the picture above, pink arrows are grouped by their colors. Each group of pink arrows indicates the change in area between each two adjacent slices.

On the other hand, every single pink arrow can also be regarded as the partial derivative $\frac\partial{\partial y}f$ at each point $(x, y)$.

Thus, summing up a group of partial derivatives means measuring the difference between two areas.

Answer 2

You could see it as an alternative version of Schwarz's theorem. When you have a map $f : \mathbb{R}^2 \to \mathbb{R}$ you would like to get to the same value if you first walk along $x$ and then $y$ as well as the other way around $y$ and then $x$.