Calculate the derivative. $\frac { d } { d x } ( [ \ln | x | ] [ \cot ( 6 x + 5 ) ] )$
A) $\frac { \cot ( 6 x - 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x - 5 )$
B) $\frac { \cot ( 6 x + 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x + 5 )$
C) $- 6 \ln | x | \csc ^ { 2 } ( 6 x + 5 )$
D) $\frac { \cot ( 6 x + 5 ) } { x }$
E) $\frac { \tan ( 6 x + 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x - 5 )$

Question

Calculate the derivative.  $\frac { d } { d x } ( [ \ln | x | ] [ \cot ( 6 x + 5 ) ] )$  
A) $\frac { \cot ( 6 x - 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x - 5 )$
B) $\frac { \cot ( 6 x + 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x + 5 )$
C) $- 6 \ln | x | \csc ^ { 2 } ( 6 x + 5 )$
D) $\frac { \cot ( 6 x + 5 ) } { x }$
E) $\frac { \tan ( 6 x + 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x - 5 )$

Quizplus · Accepted Answer

To find the derivative of the given function $\frac { d } { d x } ( [ \ln | x | ] [ \cot ( 6 x + 5 ) ] )$, we use the product rule: $(f \cdot g)' = f' \cdot g + f \cdot g'$. Here, $f = \ln | x |$ and $g = \cot ( 6 x + 5 )$. The derivative of $f = \ln | x |$ is $f' = \frac{1}{x}$, and the derivative of $g = \cot ( 6 x + 5 )$ is $g' = -6 \csc^2(6x + 5)$ because the derivative of $\cot u$ is $-\csc^2 u \cdot u'$, where $u = 6x + 5$ and $u' = 6$. Therefore, the derivative of the given function is $\frac { \cot ( 6 x + 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x + 5 )$.

Calculate the Derivative ddx([ln⁡∣x∣][cot⁡(6x+5)])\frac { d } { d x } ( [ \ln | x | ] [ \cot ( 6 x + 5 ) ] )dxd​([ln∣x∣][cot(6x+5)])

Calculate the Derivative $\frac { d } { d x } ( [ \ln | x | ] [ \cot ( 6 x + 5 ) ] )$