Recall:

If $y=g(x)*h(x)$ then $\frac{dy}{dx}=g^{\text{'}}(x)*h(x)g(x)*h(x),[$product rule$]$ If $y=\frac{g(x)}{h(x)}$ then $\frac{dy}{dx}=\frac{h(x)*g^{\text{'}}(x)-g(x)*h^{\text{'}}(x)}{(h(x){)}^2}\{$quotient rale$\}$ If $y=f(g(x))$ then $\frac{dy}{dx}=f^{\text{'}}(g(x))*g^{\text{'}}(x)$

$[$chain rule$]$

Ex$.4)$ For $F$ and $G$ functions and given that:

$F(S)=1,F(S)=-3G(S)=4G(S)=2$

a$)$ If $P(x)=F(x)*G(x),$ then evaluate $P^{\text{'}}(5).$

$P^{\text{'}}(5)$

$$

b$)$ If $Q(x)=\frac{F(x)}{G(x)},$ then evaluate $Q^{\text{'}}(5).$

$Q^{\text{'}}(5)=$

Ex$.5)$ For $R$ and $Q$ functions and given that:

$Q(-1)=5,R(-1)=-4,R(5)=7$

$Q^{\text{'}}(-1)=-3,R^{\text{'}}(-1)=2,R^{\text{'}}(5)=-1$

If $f(x)=R(Q(x)),$ then evaluate $f^{\text{'}}(-1).$

$f^{\text{'}}(-1)=$