Question

Can you help me answer 6, 9 and 12 please

Answer 1

`f(z)=(3z^2)/(5z^2+7z)`

First, we can factorize z from the denominator, then we can cancel one z

`(3z^2)/(5z^2+7z)=\frac{3z^2}{(5z^{}+7)z}=\frac{3z}{5z^{}+7}`
`f(z)=\frac{3z}{5z^{}+7}`

Then applying the quotient rule

`f^(\prime)(z)=((g\mleft(z\mright))/(h(z)))^(\prime)=(h(z)g^(\prime)(z)-h^(\prime)(z)g(z))/(h^2(z))`
`\begin{gathered} g(z)=3z \\ g^(\prime)(z)=3 \end{gathered}`
`\begin{gathered} h(z)=5z+7 \\ h^(\prime)(z)=5 \end{gathered}`

`f^(\prime)(z)=(\frac{3z}{5z^{}+7})^(\prime)=((5z+7)\cdot3-5(3z))/((5z+7)^2)`
`f^(\prime)(z)=(15z+21-15z)/((5z+7)^2)=(21)/((5z+7)^2)`

the derivate is

`(21)/((5z+7)^2)`