Chain Rule

4.2 The Chain Rule

(This topic is also in Section 4.2 in Applied Calculus or Section 11.2 of Finite Mathematics and Applied Calculus)

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Before starting with the chain rule, here is a quick quiz on using the "Calculation Thought Experiment (CTE)" discussed in the preceding tutorial. Press the "summary" button on the sidebar for a quick summary (approximately halfway down the summary page). Alternatively, press here to bring up a new window with the pertinent material from the preceding tutorial.

Q Now that we are done with the preliminaries, what is the chain rule?
A Here is an example: We know that the derivative of x³ is 3x². What, then, would you say is the deriviative of something more complicated raised top the third power, for instance (2x + x^-1.4)³ ?

Q Is it not just 3(2 + 1.4x^0.4)² ?
A No. To find the correct answer, we use the chain rule.

The Chain Rule

If u is a differentiable function of x, and f is a differentiable function of u, then:

[f(u)]

f'(u)

Example

Taking f(x) = x³, we get

u³

3u²

In words:

The derivative of a quantity cubed is 3 times that (original) quantity squared, times the derivative of the quantity.

This is sometimes referred to as an example of the generalied power rule.

More Examples

1. d

dx (1+x²)³ =
3(1+x²)² d

dx (1+x²)

=
3(1+x²)²^,2x

=
6x(1+x²)²

2. d

dx 2

(x+x²)³ =
d

dx 2(x+x²)^-3

=
-6(x+x²)^-4 d

dx (x+x²)

=
-6(x+x²)^-4 (1+2x)

=
-6(1+2x)

(x+x²)⁴

Q Why is the chain rule true?
A Press here for a proof.

Here are some for you to try.

Note Use proper graphing calculator format to input your answers (spaces are ignored).

For example, input (3x^-2 + 2/x)³(x + 1) as (3*x^(-2) + 2/x)^3 * (x+1)

Q What about things other than powers -- things such as ln(x² + 4x), for instance?
A The following table shows how we apply the chain rules to a whole variety of functions, but the derivatives of these functions appear later in the book, so have patience, or follow the links below.

Original Rule

Generalized Rule
(Chain Rule)

Comments

f(x) = g(x)

f(u) = g(u)

General form of
Chain Rule

xⁿ = nx^n-1

uⁿ = nu^n-1

Generalized Power Rule

4x^-1/2 = -2x^-3/2