# Real functions/Differentiability/Rules/Section

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).

These rules are called *sum rule*, *product rule*, *quotient rule*. The following statement is called *chain rule*.

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

holds.

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.

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.