What is  \(\frac{dy}{dx}\) euqal to ?

Calculation:

Given,

The function is \( y = 3 \sin^{-1} \left( \frac{x}{3} \right) \), and we are tasked with finding the derivative of y , i.e., \( \frac{dy}{dx} \).

The general derivative of \( \sin^{-1}(u) \) is:

\( \frac{d}{dx} \sin^{-1}(u) = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \), where\( u = \frac{x}{3} \)

\( \frac{du}{dx} = \frac{1}{3} \)

Apply the chain rule to find the derivative of y :

\( \frac{dy}{dx} = 3 \times \frac{1}{\sqrt{1 - \left( \frac{x}{3} \right)^2}} \times \frac{1}{3} \)

\( \frac{dy}{dx} = \frac{1}{\sqrt{1 - \frac{x^2}{9}}} = \frac{3}{\sqrt{9 - x^2}} \)

Hence, the correct answer is Option 3.