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2.5. THE CHAIN RULE
function g, and then the result is used as the input in f . Said differently, we can write
C(x) = f (g(x)) = sin(x
2 )
and say that C is the composition of f and g. We will refer to g, the function that is first
applied to x, as the inner function, while f , the function that is applied to the result, is the
outer function.
The main question that we answer in the present section is: 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 ′ ? In the same way that the rate of change of a product
of two functions, p(x) = f (x) · g(x), depends on the behavior of both f and g, it makes
sense intuitively that the rate of change of a composite function C(x) = f (g(x)) will also
depend on some combination of f and g and their derivatives. The rule that describes
how to compute C ′ in terms of f and g and their derivatives will be called the chain rule.
But before we can learn what the chain rule says and why it works, we first need to be
comfortable decomposing composite functions so that we can correctly identify the inner
and outer functions, as we did in the example above with C(x) = sin(x 2 ).
Preview Activity 2.5. For each function given below, identify its fundamental algebraic
structure. In particular, is the given function a sum, product, quotient, or composition
of basic functions? If the function is a composition of basic functions, state a formula
for the inner function g and the outer function f so that the overall composite function
can be written in the form f (g(x)). If the function is a sum, product, or quotient of basic
functions, use the appropriate rule to determine its derivative.
(a) h(x) = tan(2 x )
(b) p(x) = 2 x tan(x)
(c) r(x) = (tan(x)) 2
(d) m(x) = e tan(x)
(e) w(x) =
√
x + tan(x)
(f) z(x) =
tan(x)
⊲⊳
The chain rule
One of the challenges of differentiating a composite function is that it often cannot be
written in an alternate algebraic form. For instance, the function C(x) = sin(x 2 ) cannot
be expanded or otherwise rewritten, so it presents no alternate approaches to taking the
2.5. THE CHAIN RULE
function g, and then the result is used as the input in f . Said differently, we can write
C(x) = f (g(x)) = sin(x
2 )
and say that C is the composition of f and g. We will refer to g, the function that is first
applied to x, as the inner function, while f , the function that is applied to the result, is the
outer function.
The main question that we answer in the present section is: 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 ′ ? In the same way that the rate of change of a product
of two functions, p(x) = f (x) · g(x), depends on the behavior of both f and g, it makes
sense intuitively that the rate of change of a composite function C(x) = f (g(x)) will also
depend on some combination of f and g and their derivatives. The rule that describes
how to compute C ′ in terms of f and g and their derivatives will be called the chain rule.
But before we can learn what the chain rule says and why it works, we first need to be
comfortable decomposing composite functions so that we can correctly identify the inner
and outer functions, as we did in the example above with C(x) = sin(x 2 ).
Preview Activity 2.5. For each function given below, identify its fundamental algebraic
structure. In particular, is the given function a sum, product, quotient, or composition
of basic functions? If the function is a composition of basic functions, state a formula
for the inner function g and the outer function f so that the overall composite function
can be written in the form f (g(x)). If the function is a sum, product, or quotient of basic
functions, use the appropriate rule to determine its derivative.
(a) h(x) = tan(2 x )
(b) p(x) = 2 x tan(x)
(c) r(x) = (tan(x)) 2
(d) m(x) = e tan(x)
(e) w(x) =
√
x + tan(x)
(f) z(x) =
tan(x)
⊲⊳
The chain rule
One of the challenges of differentiating a composite function is that it often cannot be
written in an alternate algebraic form. For instance, the function C(x) = sin(x 2 ) cannot
be expanded or otherwise rewritten, so it presents no alternate approaches to taking the