If we have the function Y=X^3,
Y can be viewed as the volume of a cube with sides of length X.
As we increase X, the volume of the cube increases.
Suppose we increase X by just a very tiny amount, which we will call dX.
This will cause the volume of the cube to increase in all three dimensions.
The volume of the cube will increase by approximately
the volume of these three green rectangles.
Each of these rectangles has a length and a width equal to X,
and a height equal to dX.
The volume of each of these rectangles is
length multiplied by width multiplied by height,
which is X multiplied by X multiplied by dX, which we write as (X^2)dX.
The sum of the volume of all three rectangles is therefore 3 multiplied by (X^2)dX
This means that if X increases by the small increment dX,
the volume of the cube will increase by approximately 3(X^2)dX.
We give a name to this increase in volume, and we call it dY.
dY divided by dX is what we call the derivative.
Therefore, if our function is Y=X^3, then our derivative is 3(X^2).
Now let us consider another example.
In this case,
the function can be visualized as the area of a square with sides of length X.
Suppose we increase X by a just very tiny amount, which we will call dX.
This will cause the area of the square to increase in two dimensions.
The area of the square will increase by approximately
the area of these two green rectangles.
Each of these rectangles has a length of X and a width of dX.
The area of each of these two rectangles is therefore X multiplied by dX.
The sum of the areas of these two rectangles is 2(X)dX.
Therefore, if X increases by the small increment dX,
the area of the square will increase by approximately 2(X)dX.
As before, we give a name to this increase in area, and we call it dY.
As before, the derivative is dY/dX.
This means that if our function is Y=X^2, the derivative is 2X.
If we draw cubes in higher or lower dimensions,
we can repeat this analysis and get similar results.
One last point.
The increase in area of the square here
is not exactly equal to the area of the green rectangles,
due to the fact that we did not take into account the area of this red rectangle.
However, the area of this red rectangle is dX multiplied by dX.
dX is, by definition, a very small number.
Two small numbers multiplied by each other is an even smaller number.
As the value for dX approaches zero, the area of the red rectangle becomes negligible.
The same thing is true for the value for
the red rectangles in the case of the function Y=X^3.
More information about visualizing calculus and mathematics
is available in the other videos in this channel.