Taylor polynomial in function composition

Question

I have the Taylor polynomial of a function f(x):

$$4-5x+2x^2$$

and the Taylor polynomial of a function g(x):

$$2+\frac{1}{2}x-\frac{1}{8}x^2$$

Both about $$ x=0$$

How can I calculate the Taylor polynomial (at the same point) of:

$$(gf)(x) $$ ?

Answer 1

In general, a Taylor polynomial for a function about $x=0$ is $$f(x)=f(0)+f'(0)x+\frac {f''(0)}{2!}x^2$$

So for a composite function, the Taylor polynomial will be $$(fg)(x)=(fg)(0)+(fg)'(0)x+\frac {(fg)''(0)}{2!}x^2$$

So, you get the followings,

$$(fg)(0)=f(g(0))=f(2)=2$$

$$(fg)'(0)=f'(g(0))g'(0)=f'(2)g'(0)=f'(2)*\frac12$$ (this is where @arctic tern asked you a question)

$$(fg)''(0)=f''(g(0))g'(0)+f'(g(0))g''(0)=f''(2)\frac12-f'(2)\frac14$$

Answer 2

Faa di Bruno's formula gives the derivatives of $g\circ f$ at $x$ in terms of the value and derivatives of $f$ at $x$ and of $g$ at $f(x)$. But note that if $f(0) \ne 0$ (as in your example) you need the Taylor series of $g$ about $f(0)$, not about $0$.