Find the derivative of the function. $w ( x ) = 2 \sec ( x ) \cdot \tan \left( x ^ { 2 } - 2 \right)$
A) $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } \left( x ^ { 2 } - 2 \right)$
B) $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } ( 2 x - 2 )$
C) $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } + 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } ( 2 x - 2 )$
D) $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right)$
E) $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } \left( x ^ { 2 } - 4 \right)$

Question

Find the derivative of the function.   $w ( x ) = 2 \sec ( x ) \cdot \tan \left( x ^ { 2 } - 2 \right)$  
A)  $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } \left( x ^ { 2 } - 2 \right)$
B) $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } ( 2 x - 2 )$
C) $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } + 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } ( 2 x - 2 )$
D) $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right)$
E) $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } \left( x ^ { 2 } - 4 \right)$

Quizplus · Accepted Answer

To find the derivative of $w(x) = 2\sec(x) \cdot \tan(x^2 - 2)$, we use the product rule: $(f \cdot g)' = f' \cdot g + f \cdot g'$. The derivative of $\sec(x)$ is $\sec(x)\tan(x)$, and the derivative of $\tan(x^2 - 2)$ is $2x \cdot \sec^2(x^2 - 2)$. Applying the product rule and chain rule correctly leads to the expression in choice A.

Find the Derivative of the Function w(x)=2sec⁡(x)⋅tan⁡(x2−2)w ( x ) = 2 \sec ( x ) \cdot \tan \left( x ^ { 2 } - 2 \right)w(x)=2sec(x)⋅tan(x2−2)

Find the Derivative of the Function $w ( x ) = 2 \sec ( x ) \cdot \tan \left( x ^ { 2 } - 2 \right)$