an example of tangent equation calculation, from a book, that I do not understand I am in trouble with an example of an equation of a tangent from a book.
Here's what my book is writing (in french) :

I translate it (summarizing a bit) : take a T(X,Y) point on the tangent,
the slope between M and T is $\frac{Y - f(x_{0})}{X - x_{0}}$
This slope is also the derived number on M, $f'(x_{0})$ : $\frac{Y - f(x_{0})}{X - x_{0}} = f'(x_{0})$
The relationship between the coordinates (X,Y) of T are thus
$Y - f(x_{0}) = f'(x_{0}).(X - x_{0})$
But here is the sample given, that troubles me :
What the equation of the tangent in $\mathbf{x_{0} = 2}$ to the parabole of equation $\mathbf{y = x^2}$ ?

(this drawing isn't from the author, it's mine, to figure what is $y = x^2$, and what $x_{0} = 2$ or $x_{0} = 3$ would then mean)
On $x_{0} = 3, y_{0} = 9$ ; the derivative being $y' = 2x$, we have $y_{0}' = 6$

*

*First question : why does the author assign $x_{0} = 3$
if he said he is looking for $x_{0} = 2$ the line just before?
reassigning an $x_{0}$ looks strange to me.

According to 8.7, the equation to the tangent on $x_{0} = 2$


*Here, $x_{0}$ returns to its previous assignment : $x_{0} = 2$...

to the parabole is :
$Y - 9 = 6(X - 3)$ or $Y = 6X - 9$
It's very troublesome.
Especially because when I check with M(2,4),
$y_{m} = 6x_{m} - 9$ with $x_{m}=2$ gives $y_{m} = 6 \times 2 - 9 = 12 - 9 = 3$
which is not on the curve, and I am supposed to be on the tangent equation.


*If with $y = x^2 $ M has for coordinates M(2,4),
why this tangent equation is returning me a point of coordinates (2, 3) for it?


L'auteur a fait 2 fois la même coquille. Remplace simplement ses deux "$x_{0} = 2
$" par  "$x_{0} =3
$".

The author did twice the same typo.
Only replace the two "$x_{0} = 2$"'s by "$x_{0} = 3$".
