Could anybody provide a more detailed explanation of a tangent equation in its general form?

Question

In my textbook I'm currently at the topic of a tangent line to an ellipsis and hyperbola. And there I've encountered this statement:

If a curve has an equation

$$ y = f(x) $$

then an equation of a tangent line to it, which contains the point $(x_0,y_0)$ , where $y_0 = f(x_0)$ , could be written as

$$ y - y_0 = f'(x_0)(x-x_0) $$

Could anyone explain this in more detail, please? I really did not understand author's logic.

Thank you.

Answer 1

Well you should know that derivative has an geometric meaning which coresponds to inclination.

$$m=f'(x_0)$$ and the tangent line is passing through the points $(x_0,f(x_0))$ with inclination $m$. Thus,

$$m=f'(x_0)=\dfrac{y-f(x_0)}{x-x_0}$$ we are done.