Fresh day, fresh solve. 1. Differentiate $y = \arctan\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $x$. Step 1: Use a trigonometric substitution to simplify the expression inside the $\arctan$ function. Let $x = \tan\theta$. Then $dx = \sec^2\theta\,d\theta$. Also, $\sqrt{1+x^2} = \sqrt{1+\tan^2\theta} = \sqrt{\sec^2\theta} = |\sec\theta|$. Assuming $x > 0$, we have $\theta \in (0, \pi/2)$, so $\sec\theta > 0$. Substitute $