Real functions/Differentiability/Rules/Section
These rules are called sum rule, product rule, quotient rule. The following statement is called chain rule.
Let be a subset, a point, and
functions which are differentiable
in . Then the following rules for differentiability holds.- The sum is differentiable in , with
- The product is differentiable in , with
- For
,
also is differentiable in , with
- If has no zero in , then is differentiable in , with
- If has no zero in , then is differentiable in , with
(1). We write and respectively with the objects which were formulated in fact, that is
and
Summing up yields
Here, the sum is again continuous in , with value .
(2). We start again with
and
and multiply both equations. This yields
Due to
fact
for
limits,
the expression consisting of the last six summands is a continuous function, with value for
.
(3) follows from (2), since a constant function is differentiable with derivative .
(4). We have
Since is continuous in , due to
fact,
the left-hand factor converges for to , and because of the differentiability of in , the right-hand factor converges to .
(5) follows from (2) and (4).
Let denote subsets, and let
and
be functions with . Suppose that is differentiable in and that is differentiable in . Then also the composition
is differentiable in , and its derivative is
Let denote intervals, and let
be a bijective continuous function, with the inverse function
differentiable in with . Then also the inverse function is differentiable in , and
We consider the difference quotient
and have to show that the limit for exists, and obtains the value claimed. For this, let denote a sequence in , converging to . Because of fact, the function is continuous. Therefore, also the sequence with the members converges to . Because of bijectivity, for all . Thus
where the right-hand side exists, due to the condition, and the second equation follows from fact (5).
The function
is the inverse function of the function , given by (restricted to ). The derivative of in a point is . Due to fact, for , the relation
holds. In the zero point, however, is not differentiable.
The function
is the inverse function of the function , given by . The derivative of in is , which is different from for . Due to fact, we have for the relation
In the zero point, however, is not differentiable.