Consider the function given by . We are used to wring such things as:

i)

ii)

iii) . For example we would write for example.

Equally well, of course, it would be true to write .

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

The meaning of can be more devious.

It can simply be taken as synonymous with . That is .

When such is the intention it would be indisputable that means .

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

Now consider substituting in to define a function F defined as .

The chain rule says

where here means .

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

.

Here does not mean which is after all .

To see the chain rule in a more precise and unambiguous form think of as defining a function g given by , then and we see the chain rule as saying

Of course the u here is an entirely dummy symbol.