Question #40953

This question tests the student's ability to navigate the chain rule in conjunction with the branch cuts of inverse trigonometric functions.

A common lethal mistake is blindly applying the identities $\arcsin(\sin \theta) = \theta$ and $\arccos(\cos \theta) = \theta$, which incorrectly yields a derivative of $5$.

By evaluating the arguments at $x = \pi/2$, the sum-difference formulas push the angles into the third and fourth quadrants respectively.

Students must apply the ASTC quadrant sign convention to select the correct branches: $\pi - \theta$ for arcsin and $2\pi - \theta$ for arccos, which flips the signs of both derivatives, resulting in $-5$.