Question

find f(x). f(x)=\frac{ln x}{17 + x} f(x)=square (type an exact answer.)

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = \ln x$ and $v=17 + x$.

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

The derivative of $u=\ln x$ with respect to $x$ is $u'=\frac{1}{x}$, and the derivative of $v = 17+x$ with respect to $x$ is $v' = 1$.

Step3: Apply the quotient - rule

Substitute $u$, $u'$, $v$, and $v'$ into the quotient - rule formula:
\[

$$\begin{align*} f'(x)&=\frac{\frac{1}{x}(17 + x)-\ln x\times1}{(17 + x)^{2}}\\ &=\frac{\frac{17}{x}+1-\ln x}{(17 + x)^{2}}\\ &=\frac{17 + x-x\ln x}{x(17 + x)^{2}} \end{align*}$$

\]

Answer:

$\frac{17 + x-x\ln x}{x(17 + x)^{2}}$