 Math course on the derivative of a function.

This lesson involves the following concepts:

– definition of the derivative at a point;

– graphical aspect of the derivative;

– rate of increase;

– derivative of a usual function;

– derivative of a sum;

– derivative of a product;

– derivative of a quotient.

This math course was written by a teacher of national education.

## I. Derivative number and derivative of a function

f is a function defined on an interval I.

The curve (C) below is the graphical representation of f in an orthonormal frame .

M and N are two points of (C) with respective abscissas and where . Definition 1

If f is a function defined on an interval I and if .
When there is a real number d such that, for any real h close to 0, we have: We say that the function f is derivable at a and that d = f ‘(a) is the derivative of f at a.

Definition 2

If f is a function defined on an interval I and if a I.
When there is a real number d such that, for any real x I and close to a, we have: We say that the function f is derivable at a and that d = f ‘(a) is the derivative of f at a.

## II. Derivative function on an interval I

Definition:

We say that f is derivable on an interval I when it is derivable at any point of I

Remarks on notations and “physicists’ quirks

Physicists express the difference h = x – a by the symbol (increase of the variable x in the vicinity of the point a) and the difference f(x) – f(a) by (corresponding increase between the images of x and a that they assimilate to the y ordinates).

With these notations, they then write in the neighborhood of a: .

In general, on an interval I, noting “y” the function “f”, the derivative function of y will be noted: .

Historically, the notation is due to Newton and the differential notation comes from Leibniz.

## III. Equation of the tangent and affine approximation of f near x = a

Using the data from the beginning of the lesson and the graphical illustration and assuming that the function f is derivable at a:
The tangent (MP) to the curve (C) at M of abscissa a exists.

Its directrix is m = f ‘(a).

Its equation is therefore of the form: y = mx + p, where m = f ‘(a) and its intercept p is to be calculated.
For this, it is sufficient to write that (MP) passes through M( a ; f(a) ).

So we have: .
This gives: .

So y = f ‘(a) x + f(a) – a f ‘(a) which is often written in one of the easier to remember forms: or .

Therefore, the tangent (MP) to the curve (C) at M is the graphical representation of the affine function g: Let us show that this affine function is an approximation of the function f when x is close to a.
Indeed, the ordinate of the point P of abscissa x = a + h is: .

It is also written: , i.e.: .

Now, f(a+h) = f(a) + h f ‘(a) + h (h) with .

We deduce that, when h is close to zero, we have: f(a+h) f (a) + h f ‘(a).

We can therefore conclude that, when x is close to a, the affine function is an approximation of the function.

It can even be shown, but we will admit it here, that it is the best affine approximation of f in the neighborhood of a.

## IV.the derivative of usual functions. ## V.The derivation formulas Télécharger puis imprimer cette fiche en PDF

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