Visualizing the Multivariable Chain Rule \302\251Mike May, S.J., 2006 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn with(plots): The phrase "understanding the chain rule" has at several operational meanings in a calculus class: Understanding the formula for the chain rule and and being able to apply it symbolically; Being able to give a solid argument as to why the symbolic rule makes sense, with the ability to convince another student in the class; Being able to give a formal proof that will convince the professor. This worksheet aims to develop the first two meanings of the phrase.

Following our usual mantra, to understand a concept in multi-variable calculus we start with the concept in single variable calculus. Thus we begin by looking at the chain rule in single variable calculus.

We start by defining the functions f and g with composite h. Since we don't want to use the same variable to mean two different things in the same problem, we will use v for the inbetween variable. Thus v=g(x) and y=f(v). Thus z=h(x). g := x -> x^2; f := v -> sin(v); h := x -> f(g(x)); x0 := 2; y0 := g(x0);

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From single variable calculus we recall the formula 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 Thus we take the derivatives, evaluate them appropriately, and put them in the product that is called for, and compare the result to the derivative obtained by composing the function and taking the derivative directly. This lets us verify the chain rule symbolically, dgdx := diff(g(x),x); dfdv := diff(f(v),v); dfdvEvaluatedAtgx := eval(dfdv, v=g(x)); DerivativeProduct = dfdvEvaluatedAtgx*dgdx; dhdx := diff(h(x),x);

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The material above can be cut and pasted into a new execution group to work with any functions f and g we choose to define. We also want to look at a diagram that we will use when we generalize to a chain rule in several variables. We can think of variables as points and line segments as derivatives of the functions that connect the points. The chain rule says that the longer path is obtained by multiplying the derivatives connected to the segments. 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Next we want to visualize composition of functions. Note that the green and blue curves are the graphs of f and g respectively, using x, v, and y as described above, with the extra variable held constant. The red curve plots h. g := x -> x^2: f := v -> sin(v): h := x -> f(g(x)): x0 := 2; v0 := g(x0): y0 := f(g(x0)):

IiIj

del := 2: plotg := [x,g(x), y0, x=(x0-del)..(x0+del), color=blue]: plotf := [x0, v, f(v), v=(v0-del)..(v0+del), color=green]: ploth := [x, g(x), f(g(x)), x=(x0-del)..(x0+del), color=red]: spacecurve({plotg, plotf, ploth}, thickness=3, axes=boxed, labels=["x", "v", "y"]);

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By choosing orientations of [-90,0], [0, 90], and [-90,90] we can see the projections onto the x-y, y-z, and x-z planes respectively so that the red curve looks like g(x), f(y), and h(x) respectively. spacecurve({plotg, plotf, ploth}, thickness=3, axes=boxed, labels=["x", "v", "y"], orientation=[-90,1]); spacecurve({plotg, plotf, ploth}, thickness=3, axes=boxed, labels=["x", "v", "y"], orientation=[1,90], numpoints=200); spacecurve({plotg, plotf, ploth}, thickness=3, axes=boxed, labels=["x", "v", "y"], orientation=[-90,90]);

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We next want to notice that if we zoom in enough to see that the functions become linear. Intuitively that will mean that if the chain rule works for linear functions it works in general with differentiable functions. For the mathematician, it means that the intuitive argument can be turned into a proof with careful choices of deltas and epsilons. g := x -> x^3: f := v -> sin(v): h := x -> f(g(x)): x0 := 1: v0 := g(x0): z0 := f(g(x0)): del := 1: plotg := [x,g(x), h(x0), x=(x0-del)..(x0+del), color=blue]: plotf := [x0, v, f(v), v=(v0-del)..(v0+del), color=green]: ploth := [x, g(x), f(g(x)), x=(x0-del)..(x0+del), color=red]: spacecurve({plotg, plotf, ploth}, thickness=3, axes=boxed, labels=["x", "v", "y"]);

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

del := 0.01: plotg := [x,g(x), h(x0), x=(x0-del)..(x0+del), color=blue]: plotf := [x0, v, f(v), v=(v0-del)..(v0+del), color=green]: ploth := [x, g(x), f(g(x)), x=(x0-del)..(x0+del), color=red]: spacecurve({plotg, plotf, ploth}, thickness=3, axes=boxed, labels=["x", "v", "y"], orientation = [30,40]);

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