This worksheet is designed to strengthen the conceptual
understanding of the chain rule for differentiation of functions of a
two variables, each of which is a function of a third variable.
This worksheet assumes that you already understand the chain rule
for functions of a single variable. You may find it useful
to start by reviewing the Single Variable Chain Rule Applet Worksheet.
The worksheet uses an applet that can be found
by following the link provided:
Follow the link and click on the "2 Variable Chain Rule" button to launch the applet.
1) The first problem with the chain rule in this case is that there
are a lot of functions to sort out, making the graphs cluttered and
hard to understand. Notice that in all the graphical windows each
function has a corresponding tangent of the same color. Start by
deselecting the "planes and tangents" box in the control panel to
simplify the view by only looking at the graphs of the functions.
Between the four graphs there are now curves in seven colors,
blue, light blue, and purple, red, pink, and orange, and green.
For each color, give the domain variable (t, x, or y), the range variable (x, y, or z), the windows the graph is shown in, and a description for the function in terms of z(x,y), x(t), x, y(t), y, t0, x0=x(t0), and y0=y(t0). For the initial example, x(t)=t^2, y(t)=t^3, z(x,y)=sin(3*x+2*t), t0=.7, describe each of the functions in terms of t and 0.7=t0.
2) The second issue to address is zooming and linearization. Select both the "planes and tangents" and the "surfaces and curves" boxes in the control panel. Use the "<<" button for del to zoom in by bringing del down to 0.001. Describe what happens as you zoom and how the zoomed function graphs compare to their tangents.
3) Use the ">>" button for del to zoom back out by bringing del up to 1.0. Look at "Chain Rule for x" window. Identify the curves x(t), zx(x) and zxt(t), verifying that the curves and formulas match. Find the readouts and give the values of the corresponding derivatives dxdt, dzxdx, and zxtdt. What algebraic relationship holds between these values. Test three other values of t0, and verify that this same relationship holds at all points. Note that the tangent line-circular arc-tangent line construction in the Chain Rule for x window lets us draw a triangle whose slope is zx'(x0) and whose base is x'(t0), giving a height of zx'(x0)*x'(t0).
4) Repeat exercise 3 with "Chain Rule for y."
5) Zoom back in. Find the readouts for dzxtdt, dzytdt and dztdt. Give the values of these derivatives and note the algebraic relationship between them. Scroll through several values of t0 and verify that the relationship continues to hold. Since we have zoomed in and the surface looks linear, translate the relationship into a statement about the change in z value as you move around a parallelogram in 3-space.
6) Change the definitions of the functions x(t), y(t), and z(x,y) to functions of your choice. Verify that the relationships still hold.
Return to the SLU Calculus Applet page.
Last updated By Mike May, S.J., February 11, 2006.