I. Differentiate the following functions. Do NOT simplify once you have differentiated! [4 points each] [1] \( \quad y=\arctan \left(\ln \left(\sin ^{2} x\right)\right) \) [2] \( \quad y=\left(4 x^{2}-2\right)^{10}\left(5 x^{3}-x\right)^{5} \) [3] \( y=\frac{\sin \left(x^{3}-5 x^{2}\right)}{3 x^{2}-10} \) You must use the Quotient Rule on this problem.

First, let's differentiate the first function: For \( y=\arctan \left(\ln \left(\sin ^{2} x\right)\right) \), using the chain rule, we find that the derivative is \[ y' = \frac{1}{1+ \left(\ln \left(\sin ^{2} x\right)\right)^{2}} \cdot \frac{2 \cos x}{\sin x} \cdot \frac{1}{\sin^2 x} = \frac{2 \cot x}{\sin^2 x (1 + (\ln(\sin^2 x))^2)}. \] Now, for the second function: For \( y=\left(4 x^{2}-2\right)^{10}\left(5 x^{3}-x\right)^{5} \), we apply the product rule: \[ y' = \left(4 x^{2}-2\right)^{10} \cdot 5\left(5 x^{3}-x\right)^{4} \cdot \frac{d}{dx}(5 x^{3}-x) + \left(5 x^{3}-x\right)^{5} \cdot 10 \left(4 x^{2}-2\right)^{9} \cdot \frac{d}{dx}(4 x^{2}-2). \] Next, for the third function, utilizing the quotient rule: For \( y=\frac{\sin \left(x^{3}-5 x^{2}\right)}{3 x^{2}-10} \), we know that \[ y' = \frac{(3x^2 - 10)\cos(x^3 - 5x^2)(3x^2 - 10) - \sin(x^3 - 5x^2)(6x)}{(3x^2 - 10)^2}. \] These derivatives are valuable for those delving into calculus, and noticing how different rules apply can add a layer of excitement to the learning journey!