Proof: We will prove only the first statement. The proofs of the other statements are similar. For all we have
If , then , so
Proof: Let be a generic point of . Then
This missing result will be needed in some other theorems, so I've isolated it in the following lemma.
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 derivative, , but the infinitely small differential, .
Proof: For all
. Then by the product rule
The calculation is not valid at (since is not differentiable
at , and we divided by in the calculation. However is differentiable
. Hence the
Let . Consider to be a product where and . Then we can apply the product rule twice to get
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.