Visual 2 Variable Chain Rule

An Applet for Multivariable Visual Chain Rule

This applet looks at the chain rule in the case when a function from R to R^2 is composed with a function from R^2 to R to produce a function from R to R. The standard situation starts with a curve in the x-y plane parameterized by t, then considers z as a function of x and y. In the composition we then consider z a as a function of t.

A visual approach to the chain rule goes through a number of steps.

The first step is to see that zooming in on differentiable functions makes them look linear. In particular this means that we can understand the chain rule by looking at planes. Thus, the local change in z is the sum of the change made by first changing the value of x, then by changing the value of y. This reduces the problem to the sum of two chain rule problems in single variable calculus.
The second step is to visualize a way to multiply two slopes. [Recall that the single variable rule is that (fg)'(x)=f'(g(x)*g'(x).] Since slopes are heights of triangles, we multiply by using similar triangles, one triangle having a base of 1 and a height of g'(x), the other having a base of f'(g(x)).
The third step is to notice that the rule works as we move through a variety of values of t0.

The applet has 5 windows, a control panel, a t0 variable window with a slider to control the value of t0, a 3D View window, a 3D Zoomed window, a derivatives window to put things in a plane.

The t0 window has a hot spot to control the value of t0. You can use the zoom feature to get access to a broader range of values.

The 3D View is centered around the point (x(t0), y(t0), z(x(t0)), y(t0))). The surface of the graph is given along with a tangent plane. There are also 5 curves with a color coding. The curve for z as a function of t is in green. If we hold y constant and allow x to vary the graph is either red (y=y0) or pink (y=y1). If we hold x constant and vary y the curves are blue (x=x0) and light blue (x=x1).

The 3D Zoomed view is similar, but scaled by a factor of del. With the control panel, del can be brought down form 1 to 0.001. After zooming, notice the dots at the end of the parallelogram.

The derivatives view brings things into a plane. Notice that in terms of height from the axis, the red dot and blue dot add to give the green dot.

Things to notice:

Initially the 3D and 3D Zoomed window are the same.
After zooming by running down the size of del, the surface is the same as the tangent plane.
It is instructive to look down the x-axis and up the y- axis to see the edges as if they were functions of a single variable. Notice that the heights of the red and blue dots add to give the green dot.
Using the readouts on the control panel, the chain rule formula works as the value of t0 moves through a range.

Another applet visually explores the chain rule for single varable functions.