A visual approach to the chain rule goes through a number of steps.
- The first step is to get a
visual understanding of composition of
functions in one variable. To visually see composition
functions, h(x)=f(g(x)), it is
helps to have three dimensions available. The graph of
is the parametric curve (x, g(x), 0). Similarly, the graph of
z=f(y) is the parametric curve (0, y, f(y)). Finally the graph of
z=f(g(x)) is the curve (x, g(x), f(g(x))). Looking down the axes
in turn we can see the relationship between g(x), f(y) and f(g(x)).
- The second step is to see
that zooming in on differentiable
functions makes them look linear. In particular this means
we can understand the chain rule by looking at lines.
we can then verify that the slope of the composite is the product of
the slopes of the two factors, each evaluated at an appropriate point.
- The third step is to
visualize a way to multiply two slopes. This is easiest to do in
the 3D view. The slope of a line is the ratio of the height and
the base. Notice that del y is the height of one triange and the
base of the second, so that it cancels out when we multiply
slopes. Visually, as we rotate the graph to look up the y-axis,
the del y line disappears.