Real functions/Differentiability/Introduction/Section

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In this section, we consider functions

where is a subset of the real numbers. We want to explain what it means that such a function is differentiable in a point . The intuitive idea is to look at another point , and to consider the secant, given by the two points and , and then to let " move towards “. If this limiting process makes sense, the secants tend to become a tangent. However, this process only has a precise basis, if we use the concept of the limit of a function as defined earlier.

Let be a subset, a point, and

a function. For , , the number

is called the difference quotient of for

and .

The difference quotient is the slope of the secant at the graph, running through the two points and . For , this quotient is not defined. However, a useful limit might exist for . This limit represents, in the case of existence, the slope of the tangent for in the point .

Let be a subset, a point, and

a function. We say that is differentiable in if the limit

exists. In the case of existence, this limit is called the derivative of in , written

The derivative in a point is, if it exists, an element in . Quite often one takes the difference as the parameter for this limiting process, that is, one considers

The condition translates then to , . If the Function describes a one-dimensional movement, meaning a time-dependent process on the real line, then the difference quotient is the average velocity between the (time) points and and is the instantaneous velocity in .

Let , and let

be an affine-linear function. To determine the derivative in a point , we consider the difference quotient

This is constant and equals , so that the limit of the difference quotient as tends to exists and equals as well. Hence, the derivative exists in every point and is just . The slope of the affine-linear function is also its derivative.

We consider the function

The difference quotient for and is

The limit of this, as tends to , is . The derivative of in is therefore .