Page 24 - Calc for the Clueless

The chain rule. Suppose we have a composite function y = f(u), u = u(x).

Then .

Example 10—

2
Let f(x) = (x +1) 100 .
One way is to multiply this out. We dismiss this on grounds of sanity.

2
We let u = x + 1. Then y = f(u) = u 100 .

Then

We don't want to write u each time. We will imagine what u is and use the chain rule. Try it. It only takes a little
practice.

Example 11—

Find y' if y=(x +7x + 1) .
2
4/3
3
Imagine u = x + 7x + 1. .
3
2

Example 12—

Imagine u = x + 3x - 11. y' = sec (x + 3x - 11) · (4x +3).
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4
3
2
Example 13—
Find y' if y = sin (x + 3x).
4
6
This is a double composite: a function of a function of a function. We use the chain rule twice.

Let the crazy angle = v = x + 3x.
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4
3
So dv/dx = 4x + 3. Let u = sin (x + 3x) = sin v.
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6
So du/dv = cos v. So y =u and dy/du = 6u . So...