Applying Differentiation Rules to Trigonometric Functions

If the derivative of \(y=\sin^{-1} (x^2-4)\) can be expressed as \(\frac{2x}{\sqrt{f(x)}},\) what is \(f(x)?\)

If the instantaneous rate of change of \[y=\tan^{-1} \frac{x-1}{x+1}\] at \(x=23\) is \(a,\) what is the value of \(\frac{1}{a}?\)

If \(2x = \sin^{-1} \frac{192}{208}\), the rate of change of \( y = \sin x \cos x \) can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. What is \(a+b\)?

If the derivative of \(y=\cos^{-1} (6-6x)\) can be expressed as \(\frac{6}{\sqrt{f(x)}},\) what is \(f(x)?\)