Let’s work through the problems step by step. I will extract the mathematical expressions and solve them.

Problem 1

Differentiate the given functions and simplify the expressions:

(a)
$y = \ln \left( \frac{x+2 - 2\sqrt{x+1}}{x+2 + \sqrt{x+1}} \right) - \frac{2}{\sqrt{3}} \arctan \left( \frac{1 + 2\sqrt{x+1}}{\sqrt{3}} \right)$

(b)
$y = \arctan (\sqrt{x^2 - 1}) - \frac{\ln(x + \sqrt{x^2 - 1})}{x}$

Problem 2

Differentiate the following functions:

(a)
$y = \frac{(2x^5 - 3)^2 (\arctan(5x) - \sqrt{3})^2}{(3x+2)^2 (x-2)^5}$

(b)
$y = \left(x^2 + 3\right)^{1/\cos(3x)}$

Problem 3

Compute the derivative directly using the definition:

$y = e^{7 - 2x}.$

Would you like me to start solving all these problems, or focus on one of them first?