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 aI.
When there is a real number d such that, for any real xI 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.

## V.The derivation formulas

Cette publication est également disponible en : Français (French) العربية (Arabic)

Télécharger puis imprimer cette fiche en PDF

Télécharger ou imprimer cette fiche «derivative: 1st grade math course to download in PDF.» au format PDF afin de pouvoir travailler en totale autonomie.

## D'autres fiches dans la section 1st grade math class

Télécharger nos applications gratuites Mathématiques Web avec tous les cours,exercices corrigés.

• 97
In mathematics, the exponential function is the function that is noted exp and is equal to its own derivative and takes the value 1 in 0. Moreover, it is used to model phenomena in which a constant difference on the variable leads to a constant ratio. The student will need…
• 94
This math course on probability is available for free download in pdf format. I. Calculating frequencies Definition: In a bivariate statistical series, the values are usually represented in a cross tabulation. The sums of the rows and columns of a double entry table are called the table margins. They appear…
• 93
A first grade math course on numerical functions. This lesson on functions and their variations in first grade involves the following concepts: - definition of a function; - definition of the image and the antecedent; - definition of the definition set; - representative curve of a function; - direction of…
Les dernières fiches mises à jour

Voici la liste des derniers cours et exercices ajoutés au site ou mis à jour et similaire à derivative: 1st grade math course to download in PDF. .

Mathématiques Web c'est 2 146 108 fiches de cours et d'exercices téléchargées.

Copyright © 2008 - 2023 Mathématiques Web Tous droits réservés | Mentions légales | Signaler une Erreur | Contact

.
Scroll to Top
Mathématiques Web

FREE
VIEW