To find the derivative of the function \( y = 3 \sin(-4x) \), we will use the chain rule of differentiation. Let's go through the steps in detail. ### Step-by-Step Explanation 1. **Identify the Function**: The function given is \( y = 3 \sin(-4x) \). Here, we can see that it is a composition of functions: the outer function is \( 3 \sin(u) \) where \( u = -4x \). 2. **Differentiate the Outer Function**: The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \). Therefore, when we differentiate \( 3 \sin(u) \), we apply the constant multiple rule: \[ \frac{d}{du}(3 \sin(u)) = 3 \cos(u) \] 3. **Differentiate the Inner Function**: Now we need to differentiate the inner function \( u = -4x \). The derivative of \( -4x \) with respect to \( x \) is: \[ \frac{du}{dx} = -4 \] 4. **Apply the Chain Rule**: According to the chain rule, the derivative of \( y \) with respect to \( x \) is given by: \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \] Substituting the derivatives we found: \[ \frac{dy}{dx} = 3 \cos(-4x) \cdot (-4) \] Simplifying this gives: \[ \frac{dy}{dx} = -12 \cos(-4x) \] 5. **Final Answer**: Thus, the derivative \( \frac{dy}{dx} \) is: \[ \frac{dy}{dx} = -12 \cos(-4x) \] ### Evaluating the Options Now, let's evaluate the options provided: - **Option A: \( 12x \cos(4x) \)** - This option is incorrect because it incorrectly introduces an \( x \) term and does not account for the negative sign or the correct cosine function. - **Option B: \( -12x \cos(-4x) \)** - This option is also incorrect. Similar to option A, it incorrectly introduces an \( x \) term and does not match the derivative we calculated. - **Option C: \( -12 \cos(-4x) \)** - This option is correct. It matches our derived expression for \( \frac{dy}{dx} \). - **Option D: \( 12 \sin(-4x) \)** - This option is incorrect because it does not represent the derivative of the sine function; it simply repeats the original function multiplied by 12. ### Summary of Key Points - The derivative of \( y = 3 \sin(-4x) \) is found using the chain rule. - The correct derivative is \( \frac{dy}{dx} = -12 \cos(-4x) \). - The other options either introduce incorrect terms or do not follow the rules of differentiation. ### Revision Summary - Use the chain rule for differentiating composite functions. - Remember to differentiate both the outer and inner functions. - Pay attention to signs and constants when applying the chain rule. - Always verify your final answer against the options provided.