Question

find f(x).
f(x)=\frac{ln x}{15 + x}
f(x)=square
(type an exact answer.)

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = \ln x$ and $v=15 + x$.

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 $v = 15 + x$ with respect to $x$ is $v^\prime=1$.

Step3: Apply quotient - rule

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

$$\begin{align*} f^\prime(x)&=\frac{\frac{1}{x}(15 + x)-\ln x\times1}{(15 + x)^{2}}\\ &=\frac{\frac{15}{x}+1-\ln x}{(15 + x)^{2}}\\ &=\frac{15 + x - x\ln x}{x(15 + x)^{2}} \end{align*}$$

\]

Answer:

$\frac{15 + x - x\ln x}{x(15 + x)^{2}}$