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Discovering the Chain Rule
By John Zacharias
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Given a function , and a second function , . . .
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. . . let’s find the derivative of at the point where .
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First, let’s add the line .
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Now let’s go straight up from on the x-axis.
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What are the coordinates of this point?
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Now let’s go straight to the right from the last point . . .
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What are the coordinates of this point?
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What are the coordinates of this point?
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Next, let’s go straight down from the last point . . .
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What are the coordinates of this point?
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Next, let’s go straight to the left from the last point. . .
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What are the coordinates of this point?
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We have found our first point on the graph of the composite function.
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Next let’s find a second, nearby point. . .
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Start by picking a point on the x-axis near our first point . . .
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Let’s go straight up from this point on the x-axis . . .
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What are the coordinates of this point?
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Let’s go straight to the right from the last point . . .
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What are the coordinates of this point?
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What are the coordinates of this point?
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Let’s go straight down from the last point . . .
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What are the coordinates of this point?
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Next let’s go straight to the left from the last point . . .
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What are the coordinates of this point?
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And this is our second point on the composite function.
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Now let’s figure out the slope of the line between these points.
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Let’s go straight right from our first point on the composite function . . .
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What are the coordinates of this point?
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Now let’s go straight to the right from the second point.
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What are the coordinates of this point?
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Let’s go right from the first point and down from the second point.
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Let’s focus on part of this figure.
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As h approaches 0, the slope of the secant…
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. . . approaches the slope of the tangent.
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rise run So, the rise over the run at the limit will equal
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rise run The derivative of f at x = g(a).
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rise run The run is the difference in the x coordinates.
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rise run And since rise / run we must have
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rise Let’s bring back the other elements of the picture.
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rise What is the slope of the secant of the composition function?
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rise The rise is the same.
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rise The rise is the same.
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rise The rise is the same.
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rise The rise is the same.
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rise run = h The run is
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rise run = h The slope of the secant is rise / run
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rise run = h Which in the limit is
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