To find the derivative \( \frac{dy}{dx} \) of the function \( y = 3 \cos\left(\frac{x}{3}\right) \) and evaluate it at \( x = \frac{3\pi}{2} \), we will follow these steps: ### Step 1: Differentiate the Function We start with the function: \[ y = 3 \cos\left(\frac{x}{3}\right) \] To differentiate this function, we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by: \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \] In our case: - \( f(u) = 3 \cos(u) \) where \( u = \frac{x}{3} \) - The derivative of \( f(u) \) is \( f'(u) = -3 \sin(u) \) - The derivative of \( g(x) = \frac{x}{3} \) is \( g'(x) = \frac{1}{3} \) Now, applying the chain rule: \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = -3 \sin\left(\frac{x}{3}\right) \cdot \frac{1}{3} \] This simplifies to: \[ \frac{dy}{dx} = -\sin\left(\frac{x}{3}\right) \] ### Step 2: Evaluate the Derivative at \( x = \frac{3\pi}{2} \) Now we need to evaluate \( \frac{dy}{dx} \) at \( x = \frac{3\pi}{2} \): \[ \frac{dy}{dx} \bigg|_{x = \frac{3\pi}{2}} = -\sin\left(\frac{\frac{3\pi}{2}}{3}\right) = -\sin\left(\frac{\pi}{2}\right) \] We know that: \[ \sin\left(\frac{\pi}{2}\right) = 1 \] Thus: \[ \frac{dy}{dx} \bigg|_{x = \frac{3\pi}{2}} = -1 \] ### Conclusion: Final Answer The value of \( \frac{dy}{dx} \) when \( x = \frac{3\pi}{2} \) is: \[ \boxed{-1} \] ### Explanation of Other Options - **Option A: 1** - This is incorrect because it suggests that the slope of the tangent line is positive, which contradicts our calculation that resulted in a negative slope. - **Option B: -3** - This option is incorrect as it implies a steeper negative slope than what we calculated. The derivative we found was \( -1 \), not \( -3 \). - **Option C: 2** - This option is incorrect because it suggests a positive slope, which is not consistent with the behavior of the cosine function at the given point. ### Revision Summary - To differentiate \( y = 3 \cos\left(\frac{x}{3}\right) \), use the chain rule. - The derivative is \( \frac{dy}{dx} = -\sin\left(\frac{x}{3}\right) \). - Evaluate the derivative at \( x = \frac{3\pi}{2} \) to find \( \frac{dy}{dx} = -1 \). - The correct answer is \( -1 \) (Option D).