Question

Pls help with picture #1, and #2 is the example/how to do it. Thank you :)

Answer 1

The given function is

$h(x)=\frac{f(x)}{g(x)}$

Differentiate with respect x, by using the quotient rule.

$h^{\prime}(x)=\frac{f^{\prime}(x)g(x)-g^{\prime}(x)f(x)}{(g(x))^2}$

Set x=4, we get

$h^{\prime}(4)=\frac{f^{\prime}(4)g(4)-g^{\prime}(4)f(4)}{(g(4))^2}$$\text{Substitute }f^{\prime}(4)=5,f(4)=3,g^{\prime}(4)=7\text{ and }g(4)=9,\text{ we get}$$h^{\prime}(x)=\frac{5\times9-7\times3}{(9)^2}$

$h^{\prime}(x)=\frac{45-21}{81}=\frac{24}{81}=\frac{8}{27}$

Hence the required value is

$h^{\prime}(4)=\frac{8}{27}$