## 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 The expression 3x-1 x2-ex -4 is written as
 Select one a product a quotient a sum a difference a power none of the above

 Q The expression (2x-5)-1(x2+3x-1)-1.5 is written as
 Select one a product a quotient a sum a difference a power none of the above

is written as:

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 + 1.4x0.4)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:

 ddx [f(u)] = f'(u) dudx

Example

Taking f(x) = x3, we get

 ddx u3 = 3u2 dudx

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 ddx (1+x2)
=
 3(1+x2)2,2x
=
 6x(1+x2)2
2.
d

dx
2

(x+x2)3
=
 ddx 2(x+x2)-3
=
 -6(x+x2)-4 ddx (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
 ddx (3x2-4)3
 =
Q
 ddx 4(x2+x)2
 =

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)
 ddx f(x) = g(x)
 ddx f(u) = g(u) dudx
General form of
Chain Rule
 ddx xn = nx n-1
 ddx un = nun-1 dudx
Generalized Power Rule
 ddx 4x-1/2 = -2x-3/2
 ddx 4u-1/2 = -2u-3/2 dudx
 ddx ex = ex
 ddx eu = eu dudx
See next tutorial.
 ddx sin x = cos x
 ddx sin u = cos u dudx
Take me to text on trig functions!

 Q ddx 3x - 1 x2 - x-1 5 = ?

 5 3x - 1 x2 - x-1 4 3 2x + x-2
 5 3x - 1 x2 - x-1 4
 5 3(x2-x-1)-(3x-1)(2x+x-2) (x2-x-1)2 4
 5 3x - 1 x2 - x-1 4 3(x2-x-1)-(3x-1)(2x+x-2) (x2-x-1)2

 Q ddx [(x2 - 1)3(3x + 4)-1] = ?

 -6x(x2 - 1)2(3x + 4)-2 3(x2 - 1)2(3x+4)-1 - (x2 - 1)3(3x+4)-2 6x(x2 - 1)2(3x+4)-1 - 3(x2 - 1)3(3x+4)-2 6x(3x+4)-1 - (x2 - 1)(3)-2

Now try some of the exercises in Section 4.2 in Applied Calculus or Section 11.2 of Finite Mathematics and Applied Calculus

Last Updated: March, 2007