Calculus

# 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 $f(x)?$

What is the derivative of $y=3\sin^{4} x?$

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