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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkmbWZyYWNHRiQ2KC1JI21vR0YkNjBRMCZEaWZmZXJlbnRpYWxEO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSVmb3JtR1EhRicvJSdsc3BhY2VHRkYvJSdyc3BhY2VHRkYvJShtaW5zaXplR0ZGLyUobWF4c2l6ZUdGRi1GIzYlRi4tRi82MFEifkYnRjIvRjZRJmZhbHNlRicvRjlGVS9GO0ZVL0Y9RlUvRj9GVS9GQUZVL0ZDRlVGRC9GSFEkMGVtRicvRkpGZ24vRkxRIjFGJy9GTlEpaW5maW5pdHlGJy1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRltwLyUpYmV2ZWxsZWRHRlVGUS1GXm82JVEiZkYnRmFvRmRvLUkobWZlbmNlZEdGJDYkLUYjNiQtRl5vNiVRImdGJ0Zhb0Zkby1GZHA2JC1GIzYjRl1vRjJGMi1GLzYwUSI9RidGMkZURlZGV0ZYRllGWkZlbi9GRVEmaW5maXhGJy9GSFEvdGhpY2ttYXRoc3BhY2VGJy9GSkZlcUZpbkZbby1GLDYoLUYjNiYtRiw2KEYuLUYjNiVGLkZRLUZebzYlUSJ2RidGYW9GZG9GZm9GaW9GXHBGXnBGUUZgcC1GZHA2JC1GIzYjRl9yRjItSSltcGhhbnRvbUdGJDYkLUYjNiUtRl5vNiVGRkZhb0Zkby1GIzYlRltzLUYjNiZGW3NGXW8tRi82MEZhcUYyRjVGOEY6RjxGPkZARkJGYnFGZHFGZnFGS0ZNLUZebzYlUSJhRidGYW9GZG9GW3NGW3MvJSxjb25zdHJhaW50c0dRLGhlaWdodC1vbmx5RicvRmdvUSIwRidGaW9GXHBGXnAtRi82MFEpJnZlcmJhcjtGJ0YyRjVGOC9GO0Zjb0Y8Rj5GQEZCRmJxL0ZIUTJ2ZXJ5dGhpbm1hdGhzcGFjZUYnL0ZKRmB0RktGTS1GLDYoLUZncjYkLUYjNiVGW3MtRiM2JEZbcy1GXm82JVElZih4KUYnRmFvRmRvRltzRmZzLUYjNiZGX3JGX3FGaHBGW3FGaXNGaW9GXHBGXnBGK0ZRRmhwRltx 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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aWJHNiI2JkkkUkdCRzYiJCIiISIiISQiKisrKysiISIpJCIiISIiIS1JJ0NVUlZFU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkN1Q3JSQiKysrKysqKiEjNSQiKysrKkhxKiEjNSQiKzVxYV0jKSEjNTclJCIrTDszLyoqISM1JCIrdWgqXHIqISM1JCIrJj1EdEQpISM1NyUkIittSzszKiohIzUkIitbQSxGKCohIzUkIitLcTRrIykhIzU3JSQiKyoqW0M3KiohIzUkIitFI1EhUigqISM1JCIrOEQnM0YpISM1NyUkIitLbEs7KiohIzUkIis4VDJeKCohIzUkIisjZkB3RikhIzU3JSQiK2wiMy8jKiohIzUkIis2Kj5KdyohIzUkIitKVVAlRykhIzU3JSQiKyl6KltDKiohIzUkIitFYzx2KCohIzUkIiskUj82SCkhIzU3JSQiK0o5ZEcqKiEjNSQiK2g3Qyh5KiEjNSQiK1UrJ3lIKSEjNTclJCIra0lsSyoqISM1JCIrP29KKnoqISM1JCIrUkpmLyQpISM1NyUkIisocE1uJCoqISM1JCIrM0JTNikqISM1JCIrWyc+OEopISM1NyUkIitJaiIzJSoqISM1JCIrSHhcQikqISM1JCIrSSZSIT0kKSEjNTclJCIranoqWyUqKiEjNSQiKyczLmMkKSohIzUkIitcRnZDJCkhIzU3JSQiKydmeipbKiohIzUkIislUT14JSkqISM1JCIrbSNmOUwpISM1NyUkIitINzFgKiohIzUkIitGTyUpZikqISM1JCIrWCFmIlEkKSEjNTclJCIraUc5ZCoqISM1JCIrPil5PigpKiEjNSQiK1k/JltNKSEjNTclJCIrJlxDNycqKiEjNSQiK2tSNyUpKSohIzUkIitNI1E6TikhIzU3JSQiK0doSWwqKiEjNSQiK20hemkqKSohIzUkIitvdkBlJCkhIzU3JSQiK2h4UXAqKiEjNSQiK0hUVzMqKiEjNSQiKzgrKltPKSEjNTclJCIrJVJwTSgqKiEjNSQiK2QiPjEjKiohIzUkIitIYmJyJCkhIzU3JSQiK0Y1YngqKiEjNSQiK2JUIUckKiohIzUkIit5U0B5JCkhIzU3JSQiK2dFaiIpKiohIzUkIitFIioqXCUqKiEjNSQiK0JjJ1tRKSEjNTclJCIrJEg5ZCkqKiEjNSQiK3VTP2QqKiEjNSQiK0MsXiJSKSEjNTclJCIrRWZ6KikqKiEjNSQiKy8hPiVwKiohIzUkIitXdjkpUikhIzU3JSQiK2Z2KFEqKiohIzUkIis/UmsiKSoqISM1JCIrWHl4LyUpISM1NyUkIisjPmZ6KioqISM1JCIrRCl5USoqKiEjNSQiKygpNFM2JSkhIzU3JSQiKyMzLy0rIiEiKiQiK3JCaCs1ISIqJCIrRHAsPSUpISM1NyUkIitYQWgrNSEiKiQiK2d5JD0rIiEiKiQiK0ljaUMlKSEjNTclJCIrMy8tLDUhIiokIitcVjEuNSEiKiQiK2hxQUolKSEjNTclJCIrciZHOSsiISIqJCIrUT1ILzUhIiokIishPUB5VikhIzU3JSQiK01uJD0rIiEiKiQiK0guXzA1ISIqJCIrWnpTVyUpISM1NyUkIisoKltDLTUhIiokIitAKVxuKyIhIiokIitCdCk0WCkhIzU3JSQiK2dJbC01ISIqJCIrOi4peisiISIqJCIrcSNmdlgpISM1NyUkIitCNzEuNSEiKiQiKzY9QDQ1ISIqJCIrW1A3ayUpISM1NyUkIisnUXBNKyIhIiokIis1Vlc1NSEiKiQiKz4yb3ElKSEjNTclJCIrXHYoUSsiISIqJCIrNnluNjUhIiokIitWLEJ4JSkhIzU3JSQiKzdkRy81ISIqJCIrPEIiSCwiISIqJCIrIik+eCRbKSEjNTclJCIrdlFwLzUhIiokIitFeTk5NSEiKiQiKyQ+MS5cKSEjNTclJCIrUT81MDUhIiokIitSVlE6NSEiKiQiK1NGJG9cKSEjNTclJCIrLC1eMDUhIiokIitkPWk7NSEiKiQiKyVlXkxdKSEjNTclJCIrayQ9ZisiISIqJCIrIVFneSwiISIqJCIrJG9pKTQmKSEjNTclJCIrRmxLMTUhIiokIis0Kio0PjUhIiokIisqKmZPOyYpISM1NyUkIishcE1uKyIhIiokIitXL00/NSEiKiQiKyNcaEdfKSEjNTclJCIrYEc5MjUhIiokIislKT5lQDUhIiokIitAIlwkSCYpISM1NyUkIis7NWIyNSEiKiQiK0tYI0ctIiEiKiQiK1spR2VgKSEjNTclJCIreiJmeisiISIqJCIrKDNvUy0iISIqJCIrTDFJVSYpISM1NyUkIitVdE8zNSEiKiQiK1xFSkQ1ISIqJCIrTld3WyYpISM1NyUkIiswYngzNSEiKiQiKz4jZWwtIiEiKiQiKzktQWImKSEjNTclJCIrb089NDUhIiokIispei95LSIhIiokIitJem1oJikhIzU3JSQiK0o9ZjQ1ISIqJCIrJlFfIUg1ISIqJCIrV3Y1byYpISM1NyUkIislKioqKio0NSEiKiQiKyMpNElJNSEiKiQiKzkhUlhkKSEjNS1JJ0NPTE9VUkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmSSRSR0JHNiIkIiorKysrIiEiKSQiIiEiIiEkIiIhIiIhLUkqVEhJQ0tORVNTRzYiNiMiIiQtSSpBWEVTU1RZTEVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kkQk9YRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiLUkrQVhFU0xBQkVMU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlUSJ4NiJRInY2IlEieTYiLUkrUFJPSkVDVElPTkc2IjYlJCIjSSIiISQiI1MiIiEiIiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Once we have the domain small enough that the graphs look linear we want to synchronize the domains of the three curves. We also want all three axes to use the same scale. We plot g(x) from x=x0 to x=x1=x0+del. We plot f(v) from v=v0=g(x0) to v=v1=g(x1). We plot f(g(x)) from x=x0 to x=x1. To keep notation consistent, y0=h(x0) and y1=h(x1). g := x -> x^3: f := v -> sin(v): h := x -> f(g(x)): x0 := 1: v0 := g(x0): y0 := f(g(x0)): del := 0.01: spacecurve({ [x,g(x), y0, x=x0..(x0+del), color=blue], [x0, v, f(v), v=v0..g(x0+del), color=green], [x, g(x), f(g(x)), x=x0..(x0+del), color=red]}, thickness=3, axes=boxed, labels=["x", "v", "y"], scaling=constrained);

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Now we can interpret the chain rule in terms of the graph above. Notice that the blue line plots in the plane with the fixed y value y0. Its slope is g'(x0). By the definition of derivative: 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 The green line is in the plane with the fixed x value x0. Its slope is f'(v0). By the definition of derivative: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbW9HRiQ2MFEwJkRpZmZlcmVudGlhbEQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJWZvcm1HUSFGJy8lJ2xzcGFjZUdGSC8lJ3JzcGFjZUdGSC8lKG1pbnNpemVHRkgvJShtYXhzaXplR0ZILUYxNjBRIn5GJ0Y0L0Y4USZmYWxzZUYnL0Y7RlUvRj1GVS9GP0ZVL0ZBRlUvRkNGVS9GRUZVRkYvRkpRJDBlbUYnL0ZMRmduL0ZOUSIxRicvRlBRKWluZmluaXR5RictSSNtaUdGJDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvRjVRJ2l0YWxpY0YnLUYjNiVGMEZRLUZebzYlUSJ2RidGYW9GZG8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmBwLyUpYmV2ZWxsZWRHRlVGUS1JKG1mZW5jZWRHRiQ2JC1GIzYjLUZebzYlUSN2MEYnRmFvRmRvRjQtRjE2MFEiPUYnRjRGVEZWRldGWEZZRlpGZW4vRkdRJmluZml4RicvRkpRL3RoaWNrbWF0aHNwYWNlRicvRkxGY3FGaW5GW28tSSdtdW5kZXJHRiQ2JS1GMTYwUSRsaW1GJ0Y0RjdGOkY8Rj5GQC9GQ0Zjb0ZEL0ZHUSdwcmVmaXhGJ0Zmbi9GTFEudGhpbm1hdGhzcGFjZUYnRk1GTy1GIzYlLUZebzYlUSN2MUYnRmFvRmRvLUYxNjBRKCZzcmFycjtGJ0Y0RlRGVkZXRlhGWUZaRmVuRkZGZm5GaG5GaW5GW29GanAvJSxhY2NlbnR1bmRlckdGVS1GZnA2JC1GIzYjLUYsNigtRiM2JS1GXm82JVEjeTFGJ0Zhb0Zkby1GMTYwUSomdW1pbnVzMDtGJ0Y0RlRGVkZXRlhGWUZaRmVuRmBxL0ZKUTBtZWRpdW1tYXRoc3BhY2VGJy9GTEZpc0ZpbkZbby1GXm82JVEjeTBGJ0Zhb0Zkby1GIzYlRmJyRmVzRmpwRltwRl5wRmFwRmNwRjQ= The red line is a small segment of the graph of the parametric curve (x,g(x), f(g(x))). To find the derivative h'(x0) we need to look at the changes in z and x. Thus 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=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 Which justifies the chain rule in one variable. (A technical detail to note. We are assuming that f is differentiable at v0. That means that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVEjdjFGJ0YvRjItSSNtb0dGJDYwUSgmc3JhcnI7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJWZvcm1HRi4vJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGUy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1GLDYlUSN2MEYnRi9GMkYr whenever LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjeDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYwUSgmc3JhcnI7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJWZvcm1HUSFGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUYsNiVRI3gwRidGL0Yy.)

Having refreshed our memories on how to do the chain rule when composing two functions of a single variable, we next look at the case when z=f(x,y) with x=g(t) and y=h(t). We follow the same pattern we used above, starting with a formula that we verify with test case, then moving on to a visual understanding of why the formula should work.

We start with the generalization of the picture relating functions we used above. 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There are now four variable and two paths from t to z. The chain rule tells us that the derivative of f with respect to t is obtained by adding the component obtained by taking the product on each path. Symbolically, 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 We start by verifying with an example. f := (x,y) -> x*sin(y); g := t -> t^2; h := t -> 3*t-2; k := t -> f(g(t),h(t)); dkdt := diff(k(t),t); dhdt := diff(h(t),t); dgdt := diff(g(t),t); dfdx := diff(f(x,y),x); dfdy := diff(f(x,y),y); formalChainRule := dfdx*dgdt+dfdy*dhdt; formalChainRuleEvaluated := eval(formalChainRule, {x=g(t), y=h(t)}); Since we have just done the chain rule is single variable functions it is worthwhile to create two single variable functions and reinterpret the chain rule. Define fy0(x) = f(x,y0) and fx0(y) = f(x0,y). These are the two functions obtained by respectively holding y and x constant. Then using the chain rule for single variable functions from above we have: 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 Thus, the issue remaining is to show that the change in z for small changes in t is the sum of the changes on the two slice curves.

As with the single variable case, to understand why the formula we have for the chain rule should be correct, we want to zoom in until the functions appear to be linear. We will plot the surface z=f(x,y), along with the slice curves z=fy0(x)=f(x,y0) and z=fx0(y)=f(x0,y). With the test functions, del=1 shows the functions are not linear, and del=0.01 is small enough for the functions to seem to be linear. f := (x,y) -> x*sin(y): g := t -> t^2: h := t -> 3*t-2: k := t -> f(g(t),h(t)): t0 := -1.0; x0 := g(t0); y0 := h(t0); z0 := f(x0,y0); del := 1; plotf := plot3d(f(x,y),x=x0-del..x0+del, y=y0-del..y0+del, color = green, style=patchnogrid): plotcurves := spacecurve( {[g(t), y0, f(g(t),y0), color=red], [x0, h(t), f(x0,h(t)), color=blue], [g(t), h(t), f(g(t),h(t)), color=yellow]}, t=t0-del..t0+del, thickness=3): display([plotf, plotcurves], axes=boxed); del := .01; plotf := plot3d(f(x,y),x=x0-del..x0+del, y=y0-del..y0+del, color = green, style=patchnogrid): plotcurves := spacecurve( {[g(t), y0, f(g(t),y0), color=red], [x0, h(t), f(x0,h(t)), color=blue], [g(t), h(t), f(g(t),h(t)), color=yellow]}, t=t0-del..t0+del, thickness=3): display([plotf, plotcurves], axes=boxed, orientation=[30,20]); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Again we follow the pattern from the single variable case. Once our scale is small enough for the functions to look linear, we synchronize the domains, letting both x and y vary over the range from t0 to to+del. We have also added slice curves through f(x1,y1). del := .01: t1 := t0+del: p0 := [g(t0), h(t0), f(g(t0),h(t0))]; p1 := [g(t1), h(t1), f(g(t1),h(t1))]; ["f(x0,y0)", f(g(t0),h(t0)), "f(x0,y1)", f(g(t0),h(t1)), "f(x1,y1)", f(g(t1),h(t1))]; plotf := plot3d(f(x,y),x=g(t0)..g(t1), y=h(t0)..h(t1), color = green, style=patchnogrid): plotcurves := spacecurve( {[g(t), h(t0), f(g(t),h(t0)), color=red], [g(t), h(t1), f(g(t),h(t1)), color=red], [g(t0), h(t), f(g(t0),h(t)), color=blue], [g(t1), h(t), f(g(t1),h(t)), color=blue], [g(t), h(t), f(g(t),h(t)), color=yellow]}, t=t0..t1, thickness=3): display([plotf, plotcurves], axes=boxed, orientation=[30,20], scaling=constrained); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Notice that the green surface now appears to be a parallelogram in the tangent plain. If we consider it to be a parallelogram and leave the difference to the formal proof, then it is clear that the change in z along the two red paths is the same. Similarly the change in z along the two blue paths is the same. Finally, the change in z along the yellow path is the sum of the changes in z along a blue blue edge and a red edge. This explains why the derivative is the sum of the derivatives obtained from the slice paths.

We are ready for the chain rule in the general case, where there are two many variables to be able to graph conveniently. We look at the case where w is a function of three variables, x, y, and z, each of which is a function of two variables, u and v. We want to find the derivative of w with respect to u. Following our method we make a diagram with the variables and the partials. The derivative we want will be the sum obtained from all paths from w to u. 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Thus 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 In the notation we used in the multivariable case 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The argument that this gives us the derivative we are looking for uses the arguments from the two previous cases. Each of the three summands is a product of two segments in a path. From the single variable case each product gives the derivative of a function obtained from w by holding two of {x, y, z} constant and by also holding v constant. From the multivariable case, the change in w crossing the diagonal of a cube small enough for w to be linear, is the sum of the changes in w going along the three edge pathway.

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