$$\text { Find the derivative of } y \text { with respect to } x , t , \text { or } \theta \text {, as appropriate. }$$
-$$y = \frac { \ln x } { x ^ { 7 } }$$
A) $\frac { 1 - 7 \ln x } { x ^ { 14 } }$
B) $\frac { 1 - 7 \ln x } { x ^ { 8 } }$
C) $\frac { 1 + 7 \ln x } { x ^ { 14 } }$
D) $\frac { 7 \ln x - 1 } { x ^ { 8 } }$

Question

Quizplus · Accepted Answer

To find the derivative of $y = \frac{\ln x}{x^7}$, we use the quotient rule: $\left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}$. Let $f = \ln x$ and $g = x^7$. Then, $f' = \frac{1}{x}$ and $g' = 7x^6$. Substituting these into the quotient rule formula gives $\frac{(1/x)x^7 - (\ln x)(7x^6)}{x^{14}} = \frac{x^7 - 7x^6\ln x}{x^{14}} = \frac{1 - 7\ln x}{x^8}$.

Find the derivative of y with respect to x,t, or θ, as appropriate. \text { Find the derivative of } y \text { with respect to } x , t , \text { or } \theta \text {, as appropriate. } Find the derivative of y with respect to x,t, or θ, as appropriate.

$\text { Find the derivative of } y \text { with respect to } x , t , \text { or } \theta \text {, as appropriate. }$