Discovering the Chain Rule By John Zacharias

Given a function , and a second function , . . .

. . . let’s find the derivative of at the point where .

First, let’s add the line .

Now let’s go straight up from on the x-axis.

What are the coordinates of this point?

Now let’s go straight to the right from the last point . . .

What are the coordinates of this point?

What are the coordinates of this point?

Next, let’s go straight down from the last point . . .

What are the coordinates of this point?

Next, let’s go straight to the left from the last point. . .

What are the coordinates of this point?

We have found our first point on the graph of the composite function.

Next let’s find a second, nearby point. . .

Start by picking a point on the x-axis near our first point . . .

Let’s go straight up from this point on the x-axis . . .

What are the coordinates of this point?

Let’s go straight to the right from the last point . . .

What are the coordinates of this point?

What are the coordinates of this point?

Let’s go straight down from the last point . . .

What are the coordinates of this point?

Next let’s go straight to the left from the last point . . .

What are the coordinates of this point?

And this is our second point on the composite function.

Now let’s figure out the slope of the line between these points.

Let’s go straight right from our first point on the composite function . . .

What are the coordinates of this point?

Now let’s go straight to the right from the second point.

What are the coordinates of this point?

Let’s go right from the first point and down from the second point.

Let’s focus on part of this figure.

As h approaches 0, the slope of the secant…

. . . approaches the slope of the tangent.

rise run So, the rise over the run at the limit will equal

rise run The derivative of f at x = g(a).

rise run The run is the difference in the x coordinates.

rise run And since rise / run we must have

rise Let’s bring back the other elements of the picture.

rise What is the slope of the secant of the composition function?

rise The rise is the same.

rise The rise is the same.

rise The rise is the same.

rise The rise is the same.

rise run = h The run is

rise run = h The slope of the secant is rise / run

rise run = h Which in the limit is