The chain rule

In addition to learning how to differentiate a variety of basic functions, we have also been developing our ability to use rules to differentiate certain algebraic combinations of them.

There is one more natural way to combine basic functions algebraically, 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 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, $x \longrightarrow x^2 \longrightarrow \sin(x^2)$.

In terms of the elementary functions $f$ and $g$, we observe that $x$ is the input for the function $g$, and the result is used as the input for $f$. We 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.

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, $p(x) = f(x) \cdot g(x)$, 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 is 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 $C(x) = \sin(x^2)$.

Often a composite function 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 derivative. But some composite functions can be expanded or simplified, and these provide a way to explore how the chain rule works.

It may seem that Example is too elementary to illustrate how to differentiate a composite function. Linear functions are the simplest of all functions, and composing linear functions yields another linear function. While this example does not illustrate the full complexity of a composition of nonlinear functions, at the same time we remember that any differentiable function is locally linear, and thus any function with a derivative behaves like a line when viewed up close. The fact that the derivatives of the linear functions $f$ and $g$ are multiplied to find the derivative of their composition turns out to be a key insight.

We now consider a composition involving a nonlinear function.

In Example, if we let $g(x) = 2x$ and $f(x) = \sin(x)$, we observe that $C(x) = f(g(x))$. Now, $g'(x) = 2$ and $f'(x) = \cos(x)$, so we can view the structure of $C'(x)$ as $C'(x) = 2\cos(2x) = g'(x) f'(g(x))$.

In this example, as in the example involving linear functions, we see that the derivative of the composite function $C(x) = f(g(x))$ is found by multiplying the derivatives of $f$ and $g$, but with $f'$ evaluated at $g(x)$.

It makes sense intuitively that these two quantities are involved in the rate of change of a composite function: if we ask how fast $C$ is changing at a given $x$ value, it clearly matters how fast $g$ is changing at $x$, as well as how fast $f$ is changing at the value of $g(x)$. It turns out that this structure holds for all differentiable functions Like other differentiation rules, the Chain Rule can be proved formally using the limit definition of the derivative. as is stated in the Chain Rule.

As with the product and quotient rules, it is often helpful to think verbally about what the chain rule says: If $C$ is a composite function defined by an outer function $f$ and an inner function $g$, then $C'$ is given by the derivative of the outer function evaluated at the inner function, times the derivative of the inner function.

It is helpful to identify clearly the inner function $g$ and outer function $f$, compute their derivatives individually, and then put all of the pieces together by the chain rule.

As a side note, we remark that $r(x)$ is usually written as $\tan^2(x)$. 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 joins the sum, constant multiple, product, and quotient rules in our collection of techniques for finding the derivative of a function through understanding its algebraic structure and the basic functions that constitute it. It takes practice to get comfortable applying multiple rules to differentiate a single function, but using proper notation and taking a few extra steps will help.

The chain rule now adds substantially to our ability to compute derivatives. Whether we are finding the equation of the tangent line to a curve, the instantaneous velocity of a moving particle, or the instantaneous rate of change of a certain quantity, if the function under consideration is a composition, the chain rule is often an essential tool.

As we gain more experience with differention, we will become more comfortable in simply writing down the derivative without taking multiple steps. This is particularly simple when the inner function is linear, since the derivative of a linear function is a constant.

More generally, we can think about how each basic function rule has a corresponding chain rule version. The next example demonstrates this for two familiar functions.

An excellent exercise for getting comfortable with the derivative rules is to complete Example for every basic function. That is, write down a list of all the basic functions whose derivatives you know, and list their corresponding derivatives. Then, corresponding to each basic rule, write a composite function with the inner function being an unknown function $u(x)$ and the outer function being a basic function. Finally, write the chain rule for the composite function, such as $\frac{d}{dx}[\sin(u(x))] = \cos(u(x)) \cdot u'(x)$.

A composite function is one where the input variable $x$ first passes through one function, and then the resulting output passes through another. For example, the function $h(x) = 2^{\sin(x)}$ is composite since $x \longrightarrow \sin(x) \longrightarrow 2^{\sin(x)}$. Given a composite function $C(x) = f(g(x))$ where $f$ and $g$ are differentiable functions, the chain rule tells us that $C'(x) = f'(g(x)) g'(x)$.