2.2.24 Derivatives and differentials

Problem: Find the derivatives of the functions: a) \( y=\frac{1+x^{2}}{2 \sqrt{1+2 x^{2}}} \), b) \( y=e^{a x}\left[\frac{1}{2 a}+\frac{a \cos 2 b x+2 b \sin 2 b x}{2\left(a^{2}+4 b^{2}\right)}\right] \), c) \( y=\ln \sin \frac{2 x+4}{x+1} \) d) \( y=\frac{4+x^{4}}{x^{3}} \arctan \frac{x^{2}}{2}+\frac{4}{x} \), e) \( y=\left(x^{3}+4\right)^{\tan x} \), f) \( x=\arcsin \sqrt{1-t^{2}}, \quad y=(\arccos t)^{2} \).