In this unit we learn how to differentiate a ‘function of a function’. We first explain what is
meant by this term and then learn about the Chain Rule which is the technique used to perform
2. A function of a function
Consider the expression cos x2 . Immediately we note that this is different from the straightforward
cosine function, cos x. We are finding the cosine of x2 , not simply the cosine of x. We call such
an expression a ‘function of a function’.
Suppose, in general, that we have two functions, f (x) and g(x). Then
y = f (g(x))
is a function of a function. In our case, the function f is the cosine function and the function g
is the square function. We could identify them more mathematically by saying that
g(x) = x2
f (x) = cos x
f (g(x)) = f (x2 ) = cos x2
Now let’s have a look at another example. Suppose this time that f is the square function and
g is the cosine function. That is,
f (x) = x2
g(x) = cos x
f (g(x)) = f (cos x) = (cos x)2
We often write (cos x)2 as cos2 x. So cos2 x is also a function of a function.
In the following section we learn how to differentiate such a function.
3. The chain rule
In order to differentiate a function of a function, y = f (g(x)), that is to find
, we need to do
Substitute u = g(x). This gives us
y = f (u)
Next we need to use a formula that is known as the Chain Rule.