The given function is
[tex]h(x)=\frac{f(x)}{g(x)}[/tex]
Differentiate with respect x, by using the quotient rule.
[tex]h^{\prime}(x)=\frac{f^{\prime}(x)g(x)-g^{\prime}(x)f(x)}{(g(x))^2}[/tex]
Set x=4, we get
[tex]h^{\prime}(4)=\frac{f^{\prime}(4)g(4)-g^{\prime}(4)f(4)}{(g(4))^2}[/tex][tex]\text{Substitute }f^{\prime}(4)=5,f(4)=3,g^{\prime}(4)=7\text{ and }g(4)=9,\text{ we get}[/tex][tex]h^{\prime}(x)=\frac{5\times9-7\times3}{(9)^2}[/tex]
[tex]h^{\prime}(x)=\frac{45-21}{81}=\frac{24}{81}=\frac{8}{27}[/tex]
Hence the required value is
[tex]h^{\prime}(4)=\frac{8}{27}[/tex]
