Question

find the derivative of the function (f(z)) below. (f(z)=\frac{2}{ln(9z)}) (f(z)=) submit answer next item attempt 2: 1 attempt remaining.

Explanation:

Step1: Recall the quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = 2$ and $v=\ln(9z)$.

Step2: Find $u'$ and $v'$

Since $u = 2$, then $u'=0$. Using the chain - rule, if $v=\ln(9z)$, and let $t = 9z$, then $\frac{dv}{dz}=\frac{d}{dz}\ln(9z)=\frac{d}{dt}\ln(t)\cdot\frac{dt}{dz}$. We know that $\frac{d}{dt}\ln(t)=\frac{1}{t}$ and $\frac{dt}{dz}=9$, so $v'=\frac{1}{9z}\cdot9=\frac{1}{z}$.

Step3: Apply the quotient - rule

$f'(z)=\frac{u'v - uv'}{v^{2}}=\frac{0\times\ln(9z)-2\times\frac{1}{z}}{(\ln(9z))^{2}}=-\frac{2}{z(\ln(9z))^{2}}$.

Answer:

$-\frac{2}{z(\ln(9z))^{2}}$