If I need to get a linear approximation of a nonlinear function (linearize), for example approximate the values of a nonlinear function with a tangent line about point a, the two common choices seem to be derivation to obtain the slope of the tangent line, or using the first two terms of Taylor series. It seems to me taking derivatives of even high order functions is the simpler approach, so is there any advantage in using the Taylor series method instead?
Edit: I'm interested in an answer in the context of differentiable, continuous (differential) equations.
It's the same thing: Writing $f(x)\doteq f(a)+f′(a)(x−a)$ is nothing else than truncating the Taylor development of $f$, computed at $a$, after the linear term.
Uttering the word "Taylor" indicates that you could be inclined to increase the degree of approximation by taking higher terms into account, e.g., $$f(x)=f(a)+f'(a)(x-a)+{f''(a)\over 2!}(x-a)^2+{f'''(a)\over 3!}(x-a)^3+o\bigl((x-a)^3\bigr)\qquad(x\to a)\ ,$$ depending on the size of error you are willing accept.