Compositions and the Chain Rule using
Arrow Diagrams
J. B. Thoo
Department of Mathematics
University of California
Davis, CA 95616-8633
USA
jb2@math.ucdavis.edu
November 14, 1994
(To appear in PRIMUS)
Abstract
Arrow (or tree, or branch) diagrams are sometimes introduced to freshman calculus students as a mnemonic device for applying the chain rule
to functions of several variables. We suggest a particular format for an
arrow diagram that can be introduced to students early on as a way to
internalize the notion of a composition of functions, and that later can be
used as a mnemonic device for applying the chain rule to functions both
of one and of several variables.
1
Introduction
The use of arrow (or tree, or branch) diagrams as a mnemonic device for computing partial and total derivatives of functions of several variables by the chain
rule is not new. For example, Barcellos and Stein [1] use such a diagram in
the following manner: to compute the partial derivative ∂z/∂s of a function
1
z = z(x, y) where x = x(s, t) and y = y(s, t), simply add up the contributions
from all the possible paths that connect z to s in Figure 1. This yields the
desired result, ∂z/∂s = ∂z/∂x · ∂x/∂s + ∂z/∂y · ∂y/∂s.
x
s
y
t
z
Figure 1: A tree diagram for computing the partial derivatives of z.
Such a mnemonic device can, indeed, be very useful in helping a novice student master the chain rule for functions of several variables. However, arrow diagrams can be introduced much earlier in a student’s mathematical experience—
as early as the first encounter with the notion of a composition of functions.
Further, we suggest that to be useful in this context, an arrow diagram should
begin with an independent variable and end with a dependent variable (unlike
Figure 1, which begins with the dependent variable and ends with the independent variables).
2
Composition of functions
We like to tell our students in a precalculus course, for example, to think of a
composition of functions as what one does naturally when punching a calculator
√
to evaluate, say, y(7) where y(x) = x2 + 3:
7 −→ (7)2 −→ [(7)2 ] + 3 −→
p
{[(7)2 ] + 3)},
that is,
( ) −→ (
)2 −→ [ ] + 3 −→
p
{ }.
A typical quiz problem is,
Rewrite the function y(x) =
√
x2 + 3 as a composition of simpler
functions.
2
An acceptable answer would be
y(x) :
x −→ r(x) = x2 −→ s(r) = r + 3 −→ y(s) =
√
s.
Such an arrow diagram not only helps the students internalize the concept
of a composition of functions by emphasizing how the independent variable
depends on the dependent variable, but is also well suited as a mnemonic device
for carrying out the chain rule for functions of one variable.
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The chain rule: functions of one variable
When we introduce the chain rule for the first time in a freshman calculus
course, we begin by pointing out that built into the Leibniz notation “dy/dx”
is the understanding that the “numerator” y is the dependent variable and the
“denominator” x is the independent variable; that we seek a chain starting
with the independent variable x and ending with the dependent variable y; and
that our final result should be entirely in terms of the independent variable x.
Once this point is made, then remembering how to evaluate the chain rule for
functions of one variable is easy: draw an arrow diagram for dy/dx (beginning
with the independent variable x and ending with the dependent variable y) that
describes the composition of functions; evaluate the derivative at each step of
the chain (arrow diagram); and multiply together all the derivatives along the
√
chain. For example, let y = x2 + 3. To evaluate dy/dx we use the diagram
dy
:
dx
x
dr/dx=2x
−→
r = x2
ds/dr=1
−→
s = r+3
√
dy/ds=1/(2 s)
−→
y=
√
s.
Multiplying together all the derivatives along the chain then gives the desired
result,
dy
dx
=
=
dr ds dy
1
·
·
= 2x · 1 · √
dx dr ds
2 s
x
√
.
x2 + 3
A typical quiz problem is,
Suppose that s = s(t), r = r(s), u = u(r), v = v(u), w = w(v),
x = x(w), y = y(x), and z = z(y). Write out dz/dt, dz/dx, and
dy/dr.
3
∂x/∂s
x
∂z/∂x
∂z
: s
∂s
z
∂y/∂s
y
∂z/∂y
Figure 2: Arrow diagram for the chain rule for a function of several variables.
Hopefully, the students would first write out the arrow diagram
ds/dt
dr/ds
du/dr
dv/du
dw/dv
dx/dw
dy/dx
dz/dy
t −→ s −→ r −→ u −→ v −→ w −→ x −→ y −→ z,
after which obtaining dz/dt, dz/dx, and dy/dr should be relatively easy. For
instance, to obtain dy/dr simply multiply together all the derivatives along the
segment of the chain that starts with r and ends with y.
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The chain rule: functions of several variables
If a student has already been introduced to using an arrow diagram in the
composition of functions and also in differentiating a function of one variable by
the chain rule, then using such a diagram to apply the chain rule to a function
of several variables will seem quite natural.
We return to the first example,
Compute the partial derivative ∂z/∂s of a function z = z(x, y) where
x = x(s, t) and y = y(s, t).
The notation ∂z/∂s indicates that we seek a chain that begins with the independent variable s and ends with the dependent variable z. Thus, we sketch
an arrow diagram that shows all the possible paths that begin with s and end
with z (see Figure 2). Then, as before, the desired result is obtained by adding
together the contributions from all the paths, where each contribution is simply
the product of the partial derivatives along that path.
A typical quiz problem is,
Compute the partial derivatives ∂z/∂q and ∂z/∂r of a function z =
z(x, y) where x = x(p, q), y = y(p), and p = p(q, r).
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∂z/∂x
x
∂x/∂q
∂z
: q
∂q
∂x/∂p
∂p/∂q
z
p
∂z/∂y
y
dy/dp
(a) Computing dz/dq.
∂x/∂p
∂p/∂r
∂z
: r
∂r
x
∂z/∂x
p
z
dy/dp
y
∂z/∂y
(b) Computing dz/dr.
Figure 3: Arrow diagrams for computing dz/dq and dz/dr.
If the students have been following along, then we should see arrow diagrams
resembling Figures 3(a) and 3(b), giving the desired results
∂z
∂q
∂z
∂r
=
=
∂x ∂z
∂p
·
+
·
∂q ∂x ∂q
∂p ∂x ∂z
·
·
+
∂r ∂p ∂x
∂x ∂z
∂p dy ∂z
·
+
·
·
∂p ∂x ∂q dp ∂y
∂p dy ∂z
·
·
.
∂r dp ∂y
References
[1] Barcellos, A., and Stein, S. K., 1992, Calculus and Analytic Geometry,
fifth edition (New York: McGraw-Hill).
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