[tex]f(z)=\frac{3z^2}{5z^2+7z}[/tex]
First, we can factorize z from the denominator, then we can cancel one z
[tex]\frac{3z^2}{5z^2+7z}=\frac{3z^2}{(5z^{}+7)z}=\frac{3z}{5z^{}+7}[/tex][tex]f(z)=\frac{3z}{5z^{}+7}[/tex]
Then applying the quotient rule
[tex]f^{\prime}(z)=(\frac{g\mleft(z\mright)}{h(z)})^{\prime}=\frac{h(z)g^{\prime}(z)-h^{\prime}(z)g(z)}{h^2(z)}[/tex][tex]\begin{gathered} g(z)=3z \\ g^{\prime}(z)=3 \end{gathered}[/tex][tex]\begin{gathered} h(z)=5z+7 \\ h^{\prime}(z)=5 \end{gathered}[/tex]
[tex]f^{\prime}(z)=(\frac{3z}{5z^{}+7})^{\prime}=\frac{(5z+7)\cdot3-5(3z)}{(5z+7)^2}[/tex][tex]f^{\prime}(z)=\frac{15z+21-15z}{(5z+7)^2}=\frac{21}{(5z+7)^2}[/tex]
the derivate is
[tex]\frac{21}{(5z+7)^2}[/tex]
