find f(x).\nf(x)=\frac{ln x}{14 + x}\nf(x)=square\n(type an exact answer.)

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = \ln x$ and $v=14 + x$.

Step2: Find $u^\prime$ and $v^\prime$

The derivative of $u=\ln x$ with respect to $x$ is $u^\prime=\frac{1}{x}$, and the derivative of $v = 14 + x$ with respect to $x$ is $v^\prime=1$.

Step3: Apply the quotient - rule

Substitute $u$, $v$, $u^\prime$, and $v^\prime$ into the quotient - rule formula: [ \begin{align*} f^\prime(x)&=\frac{\frac{1}{x}(14 + x)-\ln x\times1}{(14 + x)^{2}}\ &=\frac{\frac{14}{x}+1-\ln x}{(14 + x)^{2}}\ &=\frac{14 + x - x\ln x}{x(14 + x)^{2}} \end{align*} ]

Answer:

$\frac{14 + x - x\ln x}{x(14 + x)^{2}}$