The derivative of a function in a 11th grade math lesson where we will find the derivative at a point and the concrete meaning of the derivative number and the equation of the tangent at a point.
In this eleventh grade lesson, we will discuss the derivative of a sum, product and quotient. The derivative and the direction of variation of a function, as well as the derivatives of usual functions.

## I. Derivative number – derivative function – tangent to a curve.

Definition:

Let f be a function defined on an interval I. The curve (C) below is the graphical representation of f in an orthonormal frame of reference . M and N are two points of (C) with respective abscissas and where .

M and N therefore have the following coordinates: and , i.e.: .

So we have: or

The directrix of the line (MN) secant to (C) is therefore

.

If the curve (C) has at M a tangent of directing coefficient d, then when the point N approaches M, that is to say when x tends to a, or, which is the same, when h tends to 0,

the secants (MN) will reach a limit position which is that of the tangent(MP) in M to (C).

This can then be translated using the governing coefficients as:

i.e.: .

So we have: .

If we call , the function defined for and by :

.

we have :

and , which is also written: .

Conversely, if there exists a real d and a function such that, for all and , we have: with ,

we deduce that: and therefore that: .

This allows us to give the three equivalent definitions:

Definition 1:

If f is a function defined on an interval 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 in a and that is the derivative of f in a.

Definition 2:

If f is a function defined on an interval I and if .

When there is a real number d such that, for any real and close to a, we have:

We say that the function f is derivable in a and that is the derivative of f in a.

## II. Derivative function on an interval I. Derivative function of a function derivable on I

Definition:

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

When f is derivable on an interval I, the function which associates to any the derivative number of f at x is called the derivative function of f on I. This function is noted .

Graphical interpretation of the derivative number.

If f is a function defined on an interval I. If and if f is derivable at , then :The representative curve of f has a tangent at the point and the directing coefficient of this tangent is the derivative number of the function f at .

Remarks:

If the graph of f does not have a tangent at the point M with abscissa , then the function f is not derivable at a. This is the case for the absolute value function in .

The graph of a function may well have a tangent at a point without the function being derivable at that point: it is sufficient that the directing coefficient of this tangent does not exist (tangent parallel to the ordinate axis).

This is the case for the square root function in .

## III. Equation of the tangent to a curve

Definition:

If function f is derivable at a, the tangent (MP) to the curve (C) at M of abscissa exists. It has the directing coefficient .

Its equation is therefore of the form: , where and its intercept p can be calculated.

It is sufficient to write that (MP) passes through .

So we have: . This gives: .

So: which is often written in one of the forms, easier to remember:

Equation of the tangent at the point :

Definition:

or .

## IV. Sign of the derivative and direction of variation of a function

We admit without proof the following theorems:

Theorem 1:

Let f be a differentiable function on an interval I.

• If f is increasing on I, then for all , we have
• If f is decreasing on I, then for all , we have: .
• If f is constant on I, then for all , we have: .
Theorem 2:

Let f be a differentiable function on an interval I.

• If, for all , we have: , then f is increasing on I.
• If, for all , we have: , then f is decreasing on I.
• If, for all , we have: , then f is constant on I.
Theorem 3:

Let f be a differentiable function on an interval I.

• If, for all , we have: ( except perhaps at isolated points where ), then f is strictly increasing on I.
• If, for all , we have: ( except perhaps at isolated points where ), then f is strictly decreasing on I.

In particular:

Let f be a function derivable on an interval .

Ownership:
• If, for any , we have $f'(x)>\,0$, then f is strictly increasing on .
• If, for any , we have , then f is strictly decreasing on .

Examples:

1) Let the function f be defined on by .

f is derivable on and for all .

– For all , we have , so f is decreasing on .

– For all, we have , so f is increasing on .

Although , we have more precise:

– For all , we have , so f is strictly decreasing on .

– For all , we have , so f is strictly increasing on .

## V. Change of sign of the derivative and extremum of a function

We admit without proof the following theorems:

Theorem:

If f is a differentiable function on an interval I,

And if f admits a local maximum or a local minimum at different from the extremities of the interval I,

So: .

### 1. special case where f is derivable on an open interval.

Ownership:

If f is a differentiable function on an open interval I,

And if f admits a local maximum or a local minimum in ,

So: .

Ownership:

If f is a differentiable function on an open interval I,

and if and if cancels for by changing sign,

Then f(a) is a local extremum of f on I.

Examples:

1) Let the function f be defined on by . f is derivable on with .

cancels out at and by changing sign, because :

• for x belonging to , we have: so f is strictly increasing on
• .
• for x belonging to , we have : so f is strictly decreasing on.
• for x belonging to , we have : $f'(x)>0$ so f is strictly increasing on .

The function thus has a local maximum at and a local minimum at .

All this study can be summarized in the table below:

Here is a piece of the graphical representations of f and :

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