derivative under integral intuition

Solution 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.