find ( f^{prime}(x) ) if ( f(x)=\frac{ln (x)}{x^{4}} ) note: a problem like this will be fill - in - the - blank on the quiz rather than multiple choice. hint: if you use quotient rule on this one, you will need to do some creative simplifying to find the correct answer below. you may like to use the product rule on this after re - writing it as two terms multiplied together.
( \frac{1}{x}-4 ln (x) x^{-5} )
( \frac{1}{x^{2}}-4+4 ln (x) x^{-5} )
( \frac{1}{x}-4+4 ln (x) x^{-5} )
( \frac{1}{x}-4 ln (x) x^{-3} )
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Question

Accepted Answer

# Explanation:
## Step1: Apply 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, $u=\ln(x)$ and $v = x^{4}$. The derivative of $\ln(x)$ is $\frac{1}{x}$ (i.e., $u^\prime=\frac{1}{x}$), and the derivative of $x^{4}$ is $4x^{3}$ (i.e., $v^\prime = 4x^{3}$).
## Step2: Substitute into the quotient rule formula
$$
\begin{align*}
f^\prime(x)&=\frac{\frac{1}{x}\cdot x^{4}-\ln(x)\cdot4x^{3}}{(x^{4})^{2}}\
&=\frac{x^{3}-4x^{3}\ln(x)}{x^{8}}\
&=\frac{x^{3}(1 - 4\ln(x))}{x^{8}}\
&=\frac{1}{x^{5}}-\frac{4\ln(x)}{x^{5}}
\end{align*}
$$

# Answer:
$\frac{1}{x^{5}}- \frac{4\ln(x)}{x^{5}}$ (corresponds to the option $\frac{1}{x - 4}-4\frac{\ln(x)}{x - 5}$ when simplified correctly as per the quotient - rule application)

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Question

Explanation:

Step1: Apply the quotient rule

Step2: Substitute into the quotient rule formula

Answer: