Let $f ( x ) = \cot \left( e ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

A) $- \csc ^ { 2 } \left( e ^ { x } \right)$
B) $- \cot ^ { 2 } \left( e ^ { x } \right)$
C) $- e ^ { x } \cot ^ { 2 } \left( e ^ { x } \right)$
D) $- e ^ { x } \csc ^ { 2 } \left( e ^ { x } \right)$
E) $e ^ { x } \csc ^ { 2 } \left( e ^ { x } \right)$

Question

Let $f ( x ) = \cot \left( e ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

A) $- \csc ^ { 2 } \left( e ^ { x } \right)$
B) $- \cot ^ { 2 } \left( e ^ { x } \right)$
C) $- e ^ { x } \cot ^ { 2 } \left( e ^ { x } \right)$
D) $- e ^ { x } \csc ^ { 2 } \left( e ^ { x } \right)$
E) $e ^ { x } \csc ^ { 2 } \left( e ^ { x } \right)$

Quizplus · Accepted Answer

To find $f'(x)$ for $f(x) = \cot(e^x)$, we use the chain rule. The derivative of $\cot(u)$ with respect to $u$ is $-\csc^2(u)$, and the derivative of $e^x$ with respect to $x$ is $e^x$. Applying the chain rule, $f'(x) = -\csc^2(e^x) \cdot e^x$, which matches option D.

Let f(x)=cot⁡(ex)f ( x ) = \cot \left( e ^ { x } \right)f(x)=cot(ex)

Let $f ( x ) = \cot \left( e ^ { x } \right)$