Compositions and the Chain Rule using Arrow Diagrams

Transcription

1 Compositions and the Chain Rule using Arrow Diagrams J. B. Thoo Department of Mathematics Universit of California Davis, CA USA 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 appling the chain rule to functions of several variables. We suggest a particular format for an arrow diagram that can be introduced to students earl on as a wa to internalie the notion of a composition of functions, and that later can be used as a mnemonic device for appling 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 b the chain rule is not new. For eample, Barcellos and Stein [1] use such a diagram in the following manner: to compute the partial derivative / s of a function 1

2 = (, ) where = (s, t) and = (s, t), simpl add up the contributions from all the possible paths that connect to s in Figure 1. This ields the desired result, / s = / / s + / / s. s t Figure 1: A tree diagram for computing the partial derivatives of. Such a mnemonic device can, indeed, be ver 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 eperience as earl as the first encounter with the notion of a composition of functions. Further, we suggest that to be useful in this contet, 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 eample, to think of a composition of functions as what one does naturall when punching a calculator to evaluate, sa, (7) where () = 2 +3: 7 (7) 2 [(7) 2 ]+3 {[(7) 2 ]+3)}, that is, ( ) ( ) 2 [ ]+3 { }. A tpical qui problem is, Rewrite the function () = as a composition of simpler functions. 2

3 An acceptable answer would be (): r() = 2 s(r) =r +3 (s) = s. Such an arrow diagram not onl helps the students internalie the concept of a composition of functions b emphasiing how the independent variable depends on the dependent variable, but is also well suited as a mnemonic device for carring out the chain rule for functions of one variable. 3 The chain rule: functions of one variable When we introduce the chain rule for the first time in a freshman calculus course, we begin b pointing out that built into the Leibni notation d/d is the understanding that the numerator is the dependent variable and the denominator is the independent variable; that we seek a chain starting with the independent variable and ending with the dependent variable ; and that our final result should be entirel in terms of the independent variable. Once this point is made, then remembering how to evaluate the chain rule for functions of one variable is eas: draw an arrow diagram for d/d (beginning with the independent variable and ending with the dependent variable ) that describes the composition of functions; evaluate the derivative at each step of the chain (arrow diagram); and multipl together all the derivatives along the chain. For eample, let = Toevaluated/d we use the diagram d d : dr/d=2 r = 2 ds/dr=1 s = r +3 d/ds=1/(2 s) = s. Multipling together all the derivatives along the chain then gives the desired result, d d = dr d ds dr d ds = s = A tpical qui problem is, Suppose that s = s(t), r = r(s), u = u(r), v = v(u), w = w(v), = (w), = (), and = (). Write out d/dt, d/d, and d/dr. 3

4 s : s / s / s / / Figure 2: Arrow diagram for the chain rule for a function of several variables. Hopefull, the students would first write out the arrow diagram t ds/dt s dr/ds r du/dr u dv/du v dw/dv w d/dw d/d d/d, after which obtaining d/dt, d/d, andd/dr should be relativel eas. For instance, to obtain d/dr simpl multipl together all the derivatives along the segment of the chain that starts with r and ends with. 4 The chain rule: functions of several variables If a student has alread been introduced to using an arrow diagram in the composition of functions and also in differentiating a function of one variable b the chain rule, then using such a diagram to appl the chain rule to a function of several variables will seem quite natural. We return to the first eample, Compute the partial derivative / s of a function = (, ) where = (s, t) and = (s, t). The notation / s indicates that we seek a chain that begins with the independent variable s and ends with the dependent variable. Thus, we sketch an arrow diagram that shows all the possible paths that begin with s and end with (see Figure 2). Then, as before, the desired result is obtained b adding together the contributions from all the paths, where each contribution is simpl the product of the partial derivatives along that path. A tpical qui problem is, Compute the partial derivatives / q and / r of a function = (, ) where = (p, q), = (p), and p = p(q, r). 4

5 / q / q : q / p p/ q p d/dp / (a) Computing d/dq. / p / r : r p/ r p d/dp / (b) Computing d/dr. Figure 3: Arrow diagrams for computing d/dq and d/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 q r = q + p q p + p q d dp = p r p + p r d dp. References [1] Barcellos, A., andstein, S. K., 1992, Calculus and Analtic Geometr, fifth edition (New York: McGraw-Hill). 5