2.5. THE CHAIN RULE
123
derivatives individually, and then put all of the pieces together to generate the derivative
of the overall composite function. To see what we mean by this, consider the function
r(x) = (tan(x))
2 .
The function r is composite, with inner function g(x) = tan(x) and outer function
f (x) = x 2 . Organizing the key information involving f , g, and their derivatives, we have
f (x) = x 2
g(x) = tan(x)
f ′ (x) = 2x
g ′ (x) = sec 2 (x)
f ′ (g(x)) = 2 tan(x)
Applying the chain rule, which tells us that r ′ (x) = f ′ (g(x))g ′ (x), we find that for
r(x) = (tan(x)) 2 , its derivative is
r
′ (x) = 2 tan(x) sec
2 (x).
As a side note, we remark that another way to write r(x) is r(x) = tan 2 (x). Observe
that in this format, the composite nature of the function is more implicit, but this is
common notation for powers of trigonometric functions: cos 4 (x), sin
5 (x), and sec 2 (x) are
all composite functions, with the outer function a power function and the inner function a
trigonometric one.
The chain rule now substantially expands the library of functions we can differentiate,
as the following activity demonstrates.
Activity 2.13.
For each function given below, identify an inner function g and outer function f to
write the function in the form f (g(x)). Then, determine f ′ (x), g ′ (x), and f ′ (g(x)), and
finally apply the chain rule to determine the derivative of the given function.
(a) h(x) = cos(x 4 )
(b) p(x) =
tan(x)
(c) s(x) = 2 sin(x)
(d) z(x) = cot 5 (x)
(e) m(x) = (sec(x) + e x ) 9
⊳
Using multiple rules simultaneously
The chain rule now joins the sum, constant multiple, product, and quotient rules in
our collection of the different techniques for finding the derivative of a function through
123
derivatives individually, and then put all of the pieces together to generate the derivative
of the overall composite function. To see what we mean by this, consider the function
r(x) = (tan(x))
2 .
The function r is composite, with inner function g(x) = tan(x) and outer function
f (x) = x 2 . Organizing the key information involving f , g, and their derivatives, we have
f (x) = x 2
g(x) = tan(x)
f ′ (x) = 2x
g ′ (x) = sec 2 (x)
f ′ (g(x)) = 2 tan(x)
Applying the chain rule, which tells us that r ′ (x) = f ′ (g(x))g ′ (x), we find that for
r(x) = (tan(x)) 2 , its derivative is
r
′ (x) = 2 tan(x) sec
2 (x).
As a side note, we remark that another way to write r(x) is r(x) = tan 2 (x). Observe
that in this format, the composite nature of the function is more implicit, but this is
common notation for powers of trigonometric functions: cos 4 (x), sin
5 (x), and sec 2 (x) are
all composite functions, with the outer function a power function and the inner function a
trigonometric one.
The chain rule now substantially expands the library of functions we can differentiate,
as the following activity demonstrates.
Activity 2.13.
For each function given below, identify an inner function g and outer function f to
write the function in the form f (g(x)). Then, determine f ′ (x), g ′ (x), and f ′ (g(x)), and
finally apply the chain rule to determine the derivative of the given function.
(a) h(x) = cos(x 4 )
(b) p(x) =
tan(x)
(c) s(x) = 2 sin(x)
(d) z(x) = cot 5 (x)
(e) m(x) = (sec(x) + e x ) 9
⊳
Using multiple rules simultaneously
The chain rule now joins the sum, constant multiple, product, and quotient rules in
our collection of the different techniques for finding the derivative of a function through