. Here

Proof: First consider the case . For all in
domain
we have

Let be a generic sequence in domain such that . Let . Then and hence by theorem 7.10 we have and hence

This proves the theorem in the case . If then (since for other values of , is not an interior point of domain). In this case

Hence

Thus in all cases the formula holds.

for all . Then and are differentiable on , and for all

Proof: If the result is clear, so we assume . For all and all , we have

(Here I've used an identity from theorem 9.21.) Let be a generic sequence in such that . Let and let . Then so by lemma 9.34 we have . Also , and for all , so by (9.38), . Hence

and this proves formula (11.4).

The proof of (11.5) is similar.

Proof: Let
, and let
. Then

Case 1: If then represents the area of the shaded region in the figure.

so by monotonicity of area

Thus

Case 2. If we can reverse the roles of and in equation
(11.8) to get

or

In both cases it follows that

Let be a generic sequence in such that . Then , so by the squeezing rule

i.e.

Hence

We have proved that .

and such that

If is differentiable at , then is differentiable at and .

This is another assumption that is really a theorem, i.e. it can be proved. Intuitively this assumption is very plausible. It says that if two functions agree on an entire interval centered at , then their graphs have the same tangents at .

Proof: Since

it follows from the localization theorem that

To see that is not differentable at , we want to show that

does not exist. Let . Then , but and we know that does not exist. Hence does not exist, i.e., is not differentiable at .

or

This notation is used in the following way

Or:

Let . Then .

Let . Then .

The notation is due to Leibnitz, and is older than our concept of function.

Leibnitz wrote the differentiation formulas as ``
," or if
, then ``
" The notation for
derivatives
is due to Joseph Louis
Lagrange (1736-1813).
Lagrange called the
*derived function* of and it is from this that we get our word *derivative*. Leibnitz called derivatives,
*differentials*
and Newton
called them *fluxions*.

Many of the early users of the calculus thought of the derivative
as the quotient of two numbers

when was ``infinitely small''. Today ``infinitely small'' real numbers are out of fashion, but some attempts are being made to bring them back. Cf