When we have a function that is a composition of two or more functions, we could use all of the techniques we have already learned to differentiate it. However, using all of those techniques to break down a function into simpler parts that we are able to differentiate can get cumbersome. Instead, we use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

To put this rule into context, let’s take a look at an example: [latex]h(x)= \sin (x^3)[/latex]. We can think of the derivative of this function with respect to [latex]x[/latex] as the rate of change of [latex]\sin (x^3)[/latex] relative to the change in [latex]x[/latex]. Consequently, we want to know how [latex]\sin (x^3)[/latex] changes as [latex]x[/latex] changes. We can think of this event as a chain reaction: As [latex]x[/latex] changes, [latex]x^3[/latex] changes, which leads to a change in [latex]\sin (x^3)[/latex]. This chain reaction gives us hints as to what is involved in computing the derivative of [latex]\sin (x^3)[/latex]. First of all, a change in [latex]x[/latex] forcing a change in [latex]x^3[/latex] suggests that somehow the derivative of [latex]x^3[/latex] is involved. In addition, the change in [latex]x^3[/latex] forcing a change in [latex]\sin (x^3)[/latex] suggests that the derivative of [latex]\sin (u)[/latex] with respect to [latex]u[/latex], where [latex]u=x^3[/latex], is also part of the final derivative.

We can take a more formal look at the derivative of [latex]h(x)= \sin (x^3)[/latex] by setting up the limit that would give us the derivative at a specific value [latex]a[/latex] in the domain of [latex]h(x)= \sin (x^3)[/latex].

This expression does not seem particularly helpful; however, we can modify it by multiplying and dividing by the expression [latex]x^3-a^3[/latex] to obtain

From the definition of the derivative, we can see that the second factor is the derivative of [latex]x^3[/latex] at [latex]x=a[/latex]. That is,

However, it might be a little more challenging to recognize that the first term is also a derivative. We can see this by letting [latex]u=x^3[/latex] and observing that as [latex]x\to a, \, u\to a^3[/latex]:

Thus, [latex]h^{\prime}(a)= \cos (a^3) \cdot 3a^2[/latex]

In other words, if [latex]h(x)= \sin (x^3)[/latex], then [latex]h^{\prime}(x)= \cos (x^3) \cdot 3x^2[/latex]. Thus, if we think of [latex]h(x)= \sin (x^3)[/latex] as the composition [latex](f\circ g)(x)=f(g(x))[/latex] where [latex]f(x)=\sin x[/latex] and [latex]g(x)=x^3[/latex], then the derivative of [latex]h(x)= \sin (x^3)[/latex] is the product of the derivative of [latex]g(x)=x^3[/latex] and the derivative of the function [latex]f(x)= \sin x[/latex] evaluated at the function [latex]g(x)=x^3[/latex]. At this point, we anticipate that for [latex]h(x)= \sin (g(x))[/latex], it is quite likely that [latex]h^{\prime}(x)= \cos (g(x))g^{\prime}(x)[/latex]. As we determined above, this is the case for [latex]h(x)= \sin (x^3)[/latex].

Now that we have derived a special case of the chain rule, we state the general case and then apply it in a general form to other composite functions. An informal proof is provided at the end of the section.