# MA101.19 The Chain Rule – One Variable

Consider the function $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = \sin{x}$. We are used to wring such things as:

i) $f'$
ii) $f'(x)$
iii) $\displaystyle \frac{df}{dx}$.  For example we would write $f'(x) = \cos{x}$ for example.

Equally well, of course, it would be true to write $f'(u) = \cos{u}$.

The meaning of (i) and (ii) are mathematically precise. $f'$ means the derived function and $f'(x)$ means the value of $f'$ at x.

The meaning of $\displaystyle \frac{df}{dx}$ can be more devious.

It can simply be taken as synonymous with $f'(x)$. That is $\displaystyle \frac{df}{dx} = f'(x)$.

When such is the intention it would be indisputable that $\displaystyle \frac{df}{du}$ means $f'(u)$.

But there are other more shady uses as we will see.

Now consider substituting $x = u^2$ in $f(x) = \sin{x}$ to define a function F defined as $f(u) = \sin{u^2}$.

The chain rule says

\begin{aligned} \displaystyle \frac{dF}{du} &= \frac{df}{dx} \cdot \frac{dx}{du} \\&= \cos{x} \cdot 2u \\&= 2u\cos{u^2} \end{aligned}

where here $\frac{dF}{du}$ means $F'(u)$.

Note that F is not equal to f, but mathematicians frequently write the chain rule as,

$\displaystyle \frac{df}{du} = \frac{df}{dx} \cdot \frac{dx}{du}$.

Here $\frac{df}{du}$ does not mean $f'(u)$ which is after all $\cos{u}$.

To see the chain rule in a more precise and unambiguous form think of $x=u^2$ as defining a function g given by $g(u)=u^2$, then $F = f \circ g$ and we see the chain rule as saying

$(f \circ g)'(u) = f'(g(u)) \cdot g'(u)$

Of course the u here is an entirely dummy symbol.

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