Answer
a) See proof
b) $ F'''=^3C_0 f'''g+^3C_1f''g'+^3C_2f'g''+^3C_3fg'''$
$F^{iv}=^4C_0f^{iv}g+^4C_1f^{iii}g^{1}+^4C_2f^{ii}g^{ii}+^4C_3f^{1}g^{iii}+^4C_4fg^{iv}$
c) $F^n =^nC_0 f^ng+^nC_1f^{n-1}g^1+^nC_2f^{n-2}g^2+....^nC_rf^{n-r}g^r+...+^nC_nfg^n$
Work Step by Step
$$F(x)=f(x)g(x)$$
Using product rule of differentiation
$$F'(x)=f'(x)g(x)+g'(x)f(x)$$
Again using product rule
$$F''(x)=\frac{d[f'(x)g(x)]}{dx}+\frac{d[f(x)g'(x)]}{dx}=[f''(x)g(x)+f'(x)g'(x)]+[f'(x)g'(x)+f(x)g''(x)]$$
$$F''(x)=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x)$$
(b)Since $F''=f'''g+3f''g+3f'g''+fg'''$
$$ F'''=^3C_0 f'''g+^3C_1f''g'+^3C_2f'g''+^3C_3fg'''$$
$$F^{iv}=^4C_0f^{iv}g+^4C_1f^{iii}g^{1}+^4C_2f^{ii}g^{ii}+^4C_3f^{1}g^{iii}+^4C_4fg^{iv}$$
Similarly
$$F^n =^nC_0 f^ng+^nC_1f^{n-1}g^1+^nC_2f^{n-2}g^2+....+^nC_rf^{n-r}g^r+...+^nC_nfg^n$$