2.5. THE CHAIN RULE
119
2.5 The chain rule
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a composite function and how do we recognize its structure algebraically?
• Given a composite function C(x) = f (g(x)) that is built from differentiable functions f and g, how do we compute C ′ (x) in terms of f , g, f ′ , and g ′ ? What is the
statement of the Chain Rule?
Introduction
In addition to learning how to differentiate a variety of basic functions, we have also been
developing our ability to understand how to use rules to differentiate certain algebraic
combinations of them. For example, we not only know how to take the derivative of
f (x) = sin(x) and g(x) = x 2 , but now we can quickly find the derivative of each of the
following combinations of f and g:
s(x) = 3x
2 − 5 sin(x),
p(x) = x
2 sin(x), and
q(x) =
sin(x)
x 2 .
Finding s ′ uses the sum and constant multiple rules, determining p ′ requires the product
rule, and q ′ can be attained with the quotient rule. Again, we note the importance of
recognizing the algebraic structure of a given function in order to find its derivative:
s(x) = 3g(x) − 5 f (x), p(x) = g(x) · f (x), and q(x) =
f (x)
g(x) .
There is one more natural way to algebraically combine basic functions, and that is by
composing them. For instance, let’s consider the function
C(x) = sin(x
2 ),
and observe that any input x passes through a chain of functions. In particular, in the
process that defines the function C(x), x is first squared, and then the sine of the result is
taken. Using an arrow diagram,
x −→ x
2 −→ sin(x
2 ).
In terms of the elementary functions f and g, we observe that x is first input in the
119
2.5 The chain rule
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a composite function and how do we recognize its structure algebraically?
• Given a composite function C(x) = f (g(x)) that is built from differentiable functions f and g, how do we compute C ′ (x) in terms of f , g, f ′ , and g ′ ? What is the
statement of the Chain Rule?
Introduction
In addition to learning how to differentiate a variety of basic functions, we have also been
developing our ability to understand how to use rules to differentiate certain algebraic
combinations of them. For example, we not only know how to take the derivative of
f (x) = sin(x) and g(x) = x 2 , but now we can quickly find the derivative of each of the
following combinations of f and g:
s(x) = 3x
2 − 5 sin(x),
p(x) = x
2 sin(x), and
q(x) =
sin(x)
x 2 .
Finding s ′ uses the sum and constant multiple rules, determining p ′ requires the product
rule, and q ′ can be attained with the quotient rule. Again, we note the importance of
recognizing the algebraic structure of a given function in order to find its derivative:
s(x) = 3g(x) − 5 f (x), p(x) = g(x) · f (x), and q(x) =
f (x)
g(x) .
There is one more natural way to algebraically combine basic functions, and that is by
composing them. For instance, let’s consider the function
C(x) = sin(x
2 ),
and observe that any input x passes through a chain of functions. In particular, in the
process that defines the function C(x), x is first squared, and then the sine of the result is
taken. Using an arrow diagram,
x −→ x
2 −→ sin(x
2 ).
In terms of the elementary functions f and g, we observe that x is first input in the