The Taylor series expansion of a function, $f(x)$, around point "a" is
$f(x) = f(a) + \frac{x-a}{1!}\frac{\partial f(a)}{\partial x} + \frac{(x-a)^2}{2!}\frac{\partial^2 f(a)}{\partial x^2} + \ldots$
which is well known (wikipedia link).
My question is, what happens if "x" is a function of "t"? Is there a Taylor series expression, similar to that above? Would it be valid for all time?
Assuming there is a valid expansion…
What if the function "f" is a function of "x" and "t" (in this case "x" does not depend on "t")? I believe this is just a multi-variable Taylor expansion as found on the wikipedia link above. How would the multi-variable series expansion differ from that above in my question…where "x" is a function of "t"?
Best Answer
For the first question, it is a typical problem of series composition based on the chain rule.
Developed around $t=a$, you should have $$f(x(t))=f(x(a))+x'(a) f'(x(a))(t-a) +\frac{1}{2!} \left(x'(a)^2 f''(x(a))+x''(a) f'(x(a))\right)(t-a)^2+\frac{1}{3!} \left(f'''(x(a)) x'(a)^3+3 x'(a) x''(a) f''(x(a))+x'''(a) f'(x(a))\right)(t-a)^3+O\left((t-a)^4\right)$$ For the next case of $f(x(t),t)$, it starts to be really tedious, involving the partial derivatives of $f$ with respect of $x$ and $t$ but it is doable.