find $f(x)$. $f(x)=\frac{9x}{30 + ln x}$ $f(x)=square$

Explanation:

Step1: Identify the quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 9x$, $v=30+\ln x$.

Step2: Find $u^\prime$

Differentiate $u = 9x$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax)=a$ (where $a = 9$), we get $u^\prime=9$.

Step3: Find $v^\prime$

Differentiate $v = 30+\ln x$ with respect to $x$. The derivative of a constant is 0 and $\frac{d}{dx}(\ln x)=\frac{1}{x}$, so $v^\prime=\frac{1}{x}$.

Step4: Apply the quotient - rule

Substitute $u$, $u^\prime$, $v$, and $v^\prime$ into the quotient - rule formula: [ \begin{align*} f^\prime(x)&=\frac{u^\prime v - uv^\prime}{v^{2}}\ &=\frac{9(30 + \ln x)-9x\cdot\frac{1}{x}}{(30+\ln x)^{2}}\ &=\frac{270 + 9\ln x-9}{(30+\ln x)^{2}}\ &=\frac{261 + 9\ln x}{(30+\ln x)^{2}} \end{align*} ]

Answer:

$\frac{261 + 9\ln x}{(30+\ln x)^{2}}$