Suppose we have the function Y equals G multiplied by H.
This function can be visualized as the area
of the rectangle with sides of length G and H.
As the variable X changes, the area of the rectangle also changes.
Suppose we increase X by a just very tiny amount, which we will call d x.
This will cause G to change by a tiny amount, which we will call dG.
This will also cause H to change by a tiny amount, which we will call dH.
As X increases by the amount dX, the area of the rectangle
will increase by approximately the area of these two green rectangles.
The area of this green rectangle is dG multiplied by H.
The area of this other green rectangle is dH multiplied by G.
The sum of these two areas is therefore
dG multiplied by H
plus dH multiplied by G.
This sum is the amount by which Y changes, which we will call dY.
Therefore if Y equals G multiplied by H,
then when we increase X by dx,
the amount by which Y increases, which we call dy, is equal to
dG multiplied by H
plus dH multiplied by G.
dY / dX is what we call the derivative of Y with respect to X,
which we refer to as Y'
dG / dX is what we call the derivative of G with respect to X,
which we refer to as G'
dH / dX is what we call the derivative of H with respect to X,
which we refer to as H'
Therefore, if we divide both sides of this equation by dX, we get the following.
Y' = G' H + H' G
Therefore if Y = G H, then the derivative of Y, which we call Y', is equal to
G' H + H' G
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 dG multiplied by dH.
dG and dH are very small numbers.
Two small numbers multiplied by each other is an even smaller number.
As the value for dX approaches zero, both dG and dH also approach zero,
and the area of the red rectangle becomes negligible.
More information about visualizing calculus and mathematics
is available in the other videos in this channel.