let $f(2)=2,f(2)=5,h(2)=5$, and $h(2)=2$.\na. if $g(z)=f(z)cdot h(z)$, determine $g(2)$.\n$g(2)=$\nb. if…

let $f(2)=2,f(2)=5,h(2)=5$, and $h(2)=2$.\na. if $g(z)=f(z)cdot h(z)$, determine $g(2)$.\n$g(2)=$\nb. if $g(w)=f(w)/h(w)$, determine $g(2)$.\n$g(2)=$

let $f(2)=2,f(2)=5,h(2)=5$, and $h(2)=2$.\na. if $g(z)=f(z)cdot h(z)$, determine $g(2)$.\n$g(2)=$\nb. if $g(w)=f(w)/h(w)$, determine $g(2)$.\n$g(2)=$

Answer

Explanation:

Step1: Recall product - rule of differentiation

The product - rule states that if $G(z)=F(z)\cdot H(z)$, then $G^{\prime}(z)=F^{\prime}(z)H(z)+F(z)H^{\prime}(z)$.

Step2: Evaluate $G^{\prime}(2)$

Substitute $z = 2$ into the product - rule formula. We know that $F(2)=2$, $F^{\prime}(2)=5$, $H(2)=5$, and $H^{\prime}(2)=2$. So $G^{\prime}(2)=F^{\prime}(2)H(2)+F(2)H^{\prime}(2)=5\times5 + 2\times2=25 + 4=29$.

Step3: Recall quotient - rule of differentiation

The quotient - rule states that if $G(w)=\frac{F(w)}{H(w)}$, then $G^{\prime}(w)=\frac{F^{\prime}(w)H(w)-F(w)H^{\prime}(w)}{H^{2}(w)}$.

Step4: Evaluate $G^{\prime}(2)$

Substitute $w = 2$ into the quotient - rule formula. We have $F(2)=2$, $F^{\prime}(2)=5$, $H(2)=5$, and $H^{\prime}(2)=2$. So $G^{\prime}(2)=\frac{F^{\prime}(2)H(2)-F(2)H^{\prime}(2)}{H^{2}(2)}=\frac{5\times5-2\times2}{5^{2}}=\frac{25 - 4}{25}=\frac{21}{25}=0.84$.

Answer:

a. $29$ b. $0.84$