If I am doing an implicit differentiation in a curve in $\Bbb R^2$ for example, because I want to get the tangent line on a certain point... How am I suposed to know if I should differentiate it in terms of $Y$, or in terms of $X$?
Am I going to get two slopes, one that's perpendicular to the other one by changing the term that I am doing that differentiation? If so, how can I guarantee that the term choosed to differentiate will give me the slope for the tangent?!
Thanks!
This is an excellent question, and it may be worth looking up the implicit function theorem for a full answer, i.e. when can you actually solve for an explicit function of $y$ or of $x$ given some implicit function $f(x,y)=0$.
However, in general in Calculus, you have a function of the form $y(x)$ implicitly defined. Meaning you have some curve in $\mathbb{R}^2$ of the form $f(x,y)=c$, but you care about solving for the rate of change of $y$ which depends on the variable $x$, which you take to be independent.