Proof: We will prove only the first statement. The proofs of the other statements are similar. For all we have

By the sum rule for limits of functions, it follows that

i.e.

then

so

If
, then
, so

If , then , so

- a)
- b)
- c)
- d)
- e)
- f)
- g)

Proof: Let be a generic point of . Then

We know that and . If we also knew that , then by basic properties of limits we could say that

which is what we claimed.

This missing result will be needed in some other theorems, so I've isolated it in the following lemma.

Proof:

Hence by the product and sum rules for limits,

His proof is as follows:

is the difference between two successive 's; let one of these be and the other into ; then we have

the omission of the quantity which is infinitely small in comparison with the rest, for it is supposed that and are infinitely small (because the lines are understood to be continuously increasing or decreasing by very small increments throughout the series of terms), will leave .[34, page 143]Notice that for Leibniz, the important thing is not the

Proof: For all

It follows from the standard limit rules that

functions with and . Suppose and are both differentiable at , and that . Then is differentiable at , and

Then by the quotient rule

Let
. Then by the product rule

(since ).

The calculation is not valid at (since is not differentiable
at , and we divided by in the calculation. However is differentiable
at
since
, i.e.,
. Hence the
formula

is valid for all .

Let . Consider to be a product where and . Then we can apply the product rule twice to get

The domains of these functions are determined by the definition of the domain of a quotient, e.g. . Prove that

(You should memorize these formulas. Although they are easy to derive, later we will want to use them backwards; i.e., we will want to find a function whose derivative is . It is not easy to derive the formulas backwards.)

- a)
- .
- b)
- (here are constants).
- c)
- .
- d)
- .

a) Express in terms of , , , , and .

b) On the basis of your answer for part a), try to guess a formula for . Then calculate , and see whether your guess was right.