\(\begin{array}{l}
a)\quad y = \tan^23x - \cot3x^2\\
\to dy = (\tan^23x - \cot3x^2)'dx\\
\to dy = \left[2\tan3x.(\tan3x)' - \dfrac{(3x^2)'}{\sin^2(3x^2)} \right]dx\\
\to dy = \left[2\tan3x\cdot\dfrac{(3x)'}{\cos^23x} - \dfrac{6x}{\sin^2(3x^2)}\right]dx\\
\to dy = \left[\dfrac{6\tan3x}{\cos^23x} - \dfrac{6x}{\sin^2(3x^2)}\right]dx\\
b)\quad y = \sqrt{\cos^22x + 1}\\
\to dy= \left(\sqrt{\cos^22x + 1}\right)'dx\\
\to dy = \dfrac{(\cos^22x +1)'}{2\sqrt{\cos^22x + 1}}dx\\
\to dy = \dfrac{2\cos2x.(\cos2x)'}{2\sqrt{\cos^22x + 1}}dx\\
\to dy = \dfrac{\cos2x.(2x)'(-\sin2x)}{\sqrt{\cos^22x + 1}}dx\\
\to dy = \dfrac{-2\cos2x.\sin2x}{\sqrt{\cos^22x + 1}}dx\\
\to dy = - \dfrac{\sin4x}{\sqrt{\cos^22x + 1}}dx\\
\end{array}\)
