I What is the true meaning of a tangent in mathematics?

[Greg Bernhardt]

From @fresh_42's Insight
https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

Yes, it is the derivative of $y.$ But what is meant by that? Obviously we have a function $x \longmapsto y=y(x)$ and a derivative $$y'=y'(x)=\dfrac{dy}{dx}=\left. \dfrac{d}{dx}\right|_{x=a}y(x)=y(a+h)-J(h)-r(h)=y'(a) $$ It now isn't obvious at all what is meant: the function $x\longmapsto y'(x)$, the value of the slope $y'(a)$, or the linear map $J,$ the Jacobi matrix, the tangent in a way? Fact is, all of them, as needed according to the situation. I don't say we should teach tangent bundles and sections, but a little bit more accuracy would smoothen the step to calculus at college.

 

[Stephen Tashi]

In the USA, a calculus problem asking for the tangent of $f(x)$ at $x = a$ is understood to ask for the equation of a line tangent to the graph at $f(x)$ at $x = a$ so $f'(x)|_{x=a}$ is a wrong answer.