Answer
$6$
Work Step by Step
F(x) = $(1+x+x^{\frac{4}{3}}+x^{\frac{5}{3}})(\frac{3x^{5}+5x^{4}+5x+1}{8x^{9}-7x^{4}+1})$
f(x) = $1+x+x^{\frac{4}{3}}+x^{\frac{5}{3}}$
g(x) = $3x^{5}+5x^{4}+5x+1$
h(x) = $8x^{9}-7x^{4}+1$
f'(x) = $1+\frac{4}{3}x^{\frac{1}{3}}+\frac{5}{3}x^{\frac{2}{3}}$
g'(x) = $15x^{4}+20x^{3}+5$
h'(x) = $72x^{8}-28x^{3}$
f(0) = $1+0+0+0$ = $1$
g(0) = $0+0+0+1$ = $1$
h(0) = $0-0+1$ = $1$
f'(0) = $1+0+0$ = $1$
g'(0) = $0+0+5$ = $5$
h'(0) = $0-0$ = $0$
F'(0) = $f(0)[\frac{h(0)g'(0)-g(0)h'(0)}{h(0)^{2}}]+(\frac{g(0)}{h(0)})f'(0)$
F'(0) = $1(\frac{(1\times5)-(1\times0)}{1^{2}})+(\frac{1}{1})1$
F'(0) = $6$
