3.6 The chain rule  (Page 4/6)

Notice that the derivative of the composition of three functions has three parts. (Similarly, the derivative of the composition of four functions has four parts, and so on.) Also, remember, we can always work from the outside in, taking one derivative at a time.

Proof

At this point, we present a very informal proof of the chain rule. For simplicity’s sake we ignore certain issues: For example, we assume that $g\left(x\right)\ne g(a)$ for $x\ne a$ in some open interval containing $a.$ We begin by applying the limit definition of the derivative to the function $h(x)$ to obtain $h^{\prime }(a)\text{:}$

Rewriting, we obtain

Although it is clear that

it is not obvious that

To see that this is true, first recall that since g is differentiable at $a,g$ is also continuous at $a.$ Thus,

Next, make the substitution $y=g(x)$ and $b=g(a)$ and use change of variables in the limit to obtain

Finally,

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The chain rule using leibniz’s notation

As with other derivatives that we have seen, we can express the chain rule using Leibniz’s notation. This notation for the chain rule is used heavily in physics applications.

$\text{For}h\left(x\right)=f\left(g\left(x\right)\right),$ let $u=g(x)$ and $y=h\left(x\right)=g\left(u\right).$ Thus,

Consequently,