Real functions/Differentiability/Rules/Section

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An illustration of the product rule: the increment of the area is about the seize of the sum of the two products of the side length and the increment of the other side length. For the infinitesimal increment, the product of the two increments is irrelevant.

Let be a subset, a point, and

functions which are differentiable

in . Then the following rules for differentiability holds.
  1. The sum is differentiable in , with
  2. The product is differentiable in , with
  3. For , also is differentiable in , with
  4. If has no zero in , then is differentiable in , with
  5. 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


Summing up yields

Here, the sum is again continuous in , with value .
(2). We start again with


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


be functions with . Suppose that is differentiable in and that is differentiable in . Then also the composition

is differentiable in , and its derivative is

Due to fact, one can write



The remainder function

is continuous in with value .

An illustration for the derivative of the inverse function. The graph of the inverse function is the reflection of the graph at the diagonal, and the tangent behaves accordingly.

Let denote intervals, and let

be a bijective continuous function, with the inverse function

Suppose that is

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.

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.