To find the derivative \( \frac{dy}{dx} \) of the function \( y = 2x \cos(2x) - \sin(2x) \) and evaluate it at \( x = \frac{\pi}{4} \), we will follow these steps: ### Step 1: Differentiate the function We need to differentiate \( y \) with respect to \( x \). The function consists of two parts: \( 2x \cos(2x) \) and \( -\sin(2x) \). We will use the product rule for the first part and the chain rule for the second part. #### Part 1: Differentiate \( 2x \cos(2x) \) Using the product rule, which states that if \( u = 2x \) and \( v = \cos(2x) \), then: \[ \frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \] - \( u = 2x \) implies \( \frac{du}{dx} = 2 \) - \( v = \cos(2x) \) implies \( \frac{dv}{dx} = -\sin(2x) \cdot 2 = -2\sin(2x) \) (using the chain rule) Now applying the product rule: \[ \frac{d}{dx}(2x \cos(2x)) = 2x(-2\sin(2x)) + \cos(2x)(2) = -4x\sin(2x) + 2\cos(2x) \] #### Part 2: Differentiate \( -\sin(2x) \) Using the chain rule: \[ \frac{d}{dx}(-\sin(2x)) = -\cos(2x) \cdot 2 = -2\cos(2x) \] ### Step 2: Combine the derivatives Now we combine the derivatives from both parts: \[ \frac{dy}{dx} = (-4x\sin(2x) + 2\cos(2x)) - 2\cos(2x) \] This simplifies to: \[ \frac{dy}{dx} = -4x\sin(2x) + 2\cos(2x) - 2\cos(2x) = -4x\sin(2x) \] ### Step 3: Evaluate \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) Now we substitute \( x = \frac{\pi}{4} \) into the derivative: \[ \frac{dy}{dx} = -4\left(\frac{\pi}{4}\right)\sin\left(2 \cdot \frac{\pi}{4}\right) \] Calculating \( 2 \cdot \frac{\pi}{4} = \frac{\pi}{2} \), we find: \[ \sin\left(\frac{\pi}{2}\right) = 1 \] Thus, substituting this back into the derivative: \[ \frac{dy}{dx} = -4\left(\frac{\pi}{4}\right)(1) = -\pi \] ### Conclusion The value of \( \frac{dy}{dx} \) when \( x = \frac{\pi}{4} \) is \( -\pi \). ### Final Answer The correct option is **B. -π**. ### Explanation of Other Options - **A. π**: This option is incorrect because we derived \( -\pi \), not \( \pi \). - **C. π/2**: This option is incorrect as it does not match our derived value of \( -\pi \). - **D. -π/2**: This option is also incorrect because it is not equal to \( -\pi \). ### Revision Summary - Use the product rule for differentiating products of functions. - Apply the chain rule for functions within functions. - Combine derivatives carefully and simplify. - Evaluate the derivative at the specified point to find the final answer.