Composite functions are formed by combining one function with another.

To form a composite, one function is substituted into the input of a second function.

In function notation, if \( g(x) \) and \( h(x) \) are two functions, then the composite of \(g\) and \(h\) is the function \(f(x)\) defined by

\(f(x) = (g \circ h)(x) = \textcolor{NavyBlue}{g(}\textcolor{BrickRed}{h(x)}\textcolor{NavyBlue}{)}\)

where \(\textcolor{NavyBlue}{g(x)}\) is the outer function and \(\textcolor{BrickRed}{h(x)}\) is the inner function. It is read either as \(g\) composed with \(h\) or as \(g\) of \(h(x)\).

For example, the function \( f(x) = (3x^2 - 7x)^5 \) may be written as the composition of two other functions.

It can be decomposed into two separate component functions: \( g(x) = x^5 \) and \( h(x) = 3x^2 - 7x \).

Note:

The order of the composition is important.

The function, \(f(x)\), is the composition of \(g\) and \(h\), which is different than the composition of \(h\) and \(g\).

The function \( h \) composed with \( g \) would be

Consider the function given in the example, \( f(x) = (3x^2 - 7x)^5 \).

It would be very time consuming to expand, simplify, and then differentiate this expression.

Instead, we will develop a method called the chain rule to differentiate composite functions.

Observe that in the previous example, where \( f(x) = (3x^2 - 7x)^5 \), if we let \(g(u)=u^5\) and we let \(u=h(x)=3x^2-7x\), then \(f(x)=g(h(x))\).

Since we know how to differentiate \(g(u)\) and \(h(x)\) individually, the chain rule allows us to differentiate \(y=g(h(x))\).

It turns out that the derivative of the composite function, \(f(x)=g(h(x))\), is the product of the derivatives of \(g(u)\) and \(h(x)\). That is, \(f'(x)=g'(u)h'(x)\).

Consider the following derivatives as rates of change:

\(\dfrac{dy}{du}\) is the rate of change of \(y\) with respect to \(u\).

\(\dfrac{du}{dx}\) is the rate of change of \(u\) with respect to \(x\).

\(\dfrac{dy}{dx}\) is the rate of change of \(y\) with respect to \(x\).

Suppose \(y\) changes \(3\) times as fast as \(u\), and \(u\) changes \(4\) times as fast as \(x\). Then it makes sense that \(y\) changes \(3\times4=12\) times as fast as \(x\).

That is, the rate of change in \(y\) with respect to \(x\) is equal to the product of the other two rates.

Note:

Before using the chain rule, it is very important to first identify the inner and outer functions that make up the composition.