Linear Approximation of a product of functions

Question

If

$f(x) = g(x)h(x)$

does the linear approximation of $f(x)$ equals the linear approximation of $g(x)$ times the linear approximation of $h(x)$?

if it is true:

1. what is the proof?
2. is it true for quadratic approximations as well?

Answer 1

Let $F:x\mapsto ax+b$ be the linear approximation of $f$ around $0$, and $G:x\mapsto cx+d$ be the one of $g$. Then we have $FG:x\mapsto (ax+b)(cx+d) = ac x^2 +(bc+ad)x + bd$ which is clearly not linear.

But actually the linear approximation of $fg$ is $x\mapsto (bc+ad)x + bd$ since you just need to get rid of the $x^2$ term.

For quadratic approximations, it's the same : you can multiply the two approximations, but then you need to get rid of all the terms with $x^k$ where $k>4$.

So to answer your question, except for some particular cases, it is false.