I'm going through a review class on calculus and right now we are covering Taylor series. We were given a product rule, but the derivation of this rules didn't make sense to me. I can't figure out how the author came to the conclusions. Here is the rule
Product rule for Taylor polynomials
Suppose we consider the product of a function with a factor $(x-a)$, in other words $f(x)=(x-a)g(x)$. Then the $(n+1)$th order Taylor polynomial $T_{n+1}(x)$ of $f(x)$ centered at the point $x=a$ is given in terms of the $n$th order Taylor polynomial $T^g_n(x)$ of $g(x)$ as follows $$ T_{n+1}(x)=(x-a)T^g_n(x) $$
By applying this rule iteratively we also find that the $(n+k)$th order Taylor polynomial $T_{n+k}(x)$ of the function $f(x)=(x-a)^kg(x)$ centered at $x=a$ is given in terms of the $n$th order Taylor polynomial $T^g(x)$ of $g(x)$ as $$ T_{n+k}(x)=(x-a)^k T^g_n(x) $$
In fact, product rules exist for other kinds of products of function. However, the one presented above will suffice in a great many cases.
I've been at it trying to derive the rule for quite some time now, and I can't figure it out. (I think I'm just too tired) I would really appreciate a more detailed derivation of this rule.