Derivative: 1st grade math course to download in PDF.

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1 I. Derivative number and derivative of a function
2 II. Derivative function on an interval I
3 III. Equation of the tangent and affine approximation of f near x = a
4 IV.the derivative of usual functions.
5 V.The derivation formulas

Math course on the derivative of a function.

This math course on the derivative in the first year of secondary school is available for free download in PDF format.

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 $(O,\vec{i},\vec{j})$ .

M and N are two points of (C) with respective abscissas $a\in\,I$ and $x\,=\,a\,+\,h\,\in\,I$ where $h\in\,\mathbb{R}^*$ .

Definition 1

If f is a function defined on an interval I and if $a\in\,I$ .
When there is a real number d such that, for any real h close to 0, we have:

$\lim_{h\,\to\,0}\frac{f(a+h)-f(a)}{h}=d$

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 $\in$ I.
When there is a real number d such that, for any real x $\in$ I and close to a, we have:

$\lim_{x,\to,a}\frac{f(x)-f(a)}{x-a}=d$

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 $\Delta\,x$ (increase of the variable x in the vicinity of the point a) and the difference f(x) – f(a) by $\Delta\,y$ (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: $\lim_{\Delta\,x\,\to\,0}\frac{\,\Delta\,y}{\Delta\,x\,}=f'(a)$ .

In general, on an interval I, noting “y” the function “f”, the derivative function of y will be noted: $f'=\frac{dy}{dx}$

Historically, the notation $f\,'(x)$ is due to Newton and the differential notation $\frac{dy}{dx}$ 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: $f(a)\,=,f\,'(a)\,\times \,a\,+\,p$ .
This gives: $p\,=\,f(a)\,-\,a\,f\,'(a)$ .

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

$\mathbf{y\,=\,f\,'(a)\,(x-a)\,+\,f(a)}$ or $\mathbf{y\,-\,f(a)\,=\,f\,'(a)\,(x-a)}$ .

Therefore, the tangent (MP) to the curve (C) at M is the graphical representation of the affine function g:

$g:x\,\mapsto \,f'(a)(x-a)+f(a)$

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: $g(x)\,=\,f\,'(a)\,(x-a)\,+\,f(a)$ .

It is also written: $g(a\,+\,h)\,=\,f\,'(a)\,(a\,+\,h\,-\,a)\,+\,f(a)$ , i.e.: $g(a\,+\,h)\,=\,f(a)\,+\,h\,f\,'(a)$ .

Now, f(a+h) = f(a) + h f ‘(a) + h $\varphi$ (h) with $\lim_{h\,\to\,0}\varphi\,(h)=\,0$ .

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

We can therefore conclude that, when x is close to a, the affine function $g:x\,\mapsto \,f'(a)(x-a)+f(a)$ 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

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