10.6 The chain rule ‣ Chapter 10 Differential calculus ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd‣ Chapter 10 Differential calculus ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd

The chain rule is used to find the derivatives of "functions of a function". Mathematically, these are written as

\frac{d}{dx}[y(u(x))]

and it’s possible to find this just by known the derivatives of $y(u)$ and $u(x)$ .

The proof is a little involved, so for the moment you can find it at http://kruel.co/math/chainrule.pdf.

The result we’re after is that

\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}

(10.34)

What is the first thing to do when approaching a question (well after having considered that the chain rule might be relevant)? We must find the nested functions. I find it helpful to think about the parts I find hard to differentiate. For example, in our example, I know how to differentiate $\frac{1}{\text{some variable}}$ , and also $\sqrt{x}+2$ but not

f(x)=\frac{1}{\sqrt{x}+2}.

(10.36)

Therefore, it is not unreasonable to create a variable $u:=\sqrt{x}+2$ and try to use the chain rule. We can write

f(x)=\frac{1}{u}

(10.37)

This can be a little confusing, because $x$ no longer seems to appear in the function, but in reality $u$ is implicitly a function of $x$ (i.e. we know that $u$ depends on $x$ ). Applying the chain rule we know that

\displaystyle\frac{df}{dx}=\frac{d}{du}\left(\frac{1}{u}\right)\times\frac{d}{% dx}(\sqrt{x}+2)

(10.38)

We now know how to differentiate all the individual pieces. First we know that

$\displaystyle\frac{d}{du}\left(\frac{1}{u}\right)$	$\displaystyle=\frac{d}{du}\left(u^{-1}\right)$	(10.39)
	$\displaystyle=-u^{-2}$	(10.40)
	$\displaystyle=-\frac{1}{u^{2}}$	(10.41)
	$\displaystyle=-\frac{1}{(\sqrt{x}+2)^{2}}.$	(10.42)

Then we can also differentiate $\sqrt{x}+2$ with respect to $x$ , which is just

\frac{1}{2}\frac{1}{\sqrt{x}}

(10.43)

Therefore we get that overall

\frac{df}{dx}=-\frac{1}{2}\times\frac{1}{\sqrt{x}}\times\frac{1}{(\sqrt{x}+2)^% {2}}

(10.44)

10.6 The chain rule

Example 10.6.1