Find the derivative of the function. \[ g(x)=9 \arccos \left(\frac{x}{2}\right) \] \( g^{\prime}(x)=\square \)

First, let’s find the derivative of \( g(x) = 9 \arccos\left(\frac{x}{2}\right) \). We use the chain rule and the derivative of \( \arccos(u) \), which is \( -\frac{1}{\sqrt{1-u^2}} \). First, differentiate: \[ g^{\prime}(x) = 9 \cdot \left(-\frac{1}{\sqrt{1 - \left(\frac{x}{2}\right)^2}}\right) \cdot \left(\frac{d}{dx}\left(\frac{x}{2}\right)\right) \] The derivative of \( \frac{x}{2} \) is \( \frac{1}{2} \). So we have: \[ g^{\prime}(x) = 9 \cdot \left(-\frac{1}{\sqrt{1 - \frac{x^2}{4}}}\right) \cdot \frac{1}{2} \] This simplifies to: \[ g^{\prime}(x) = -\frac{9}{2\sqrt{1 - \frac{x^2}{4}}} \] Thus, the final answer is: \[ g^{\prime}(x) = -\frac{9}{2\sqrt{1 - \frac{x^2}{4}}} \]