To find the derivative of the function \( y = 2x^2(2x - 1) \) at the point \( x = -1 \), we will follow these steps: ### Step 1: Differentiate the Function We need to apply the product rule of differentiation because our function is a product of two functions: \( u = 2x^2 \) and \( v = (2x - 1) \). The product rule states that if you have two functions \( u \) and \( v \), the derivative of their product \( y = uv \) is given by: \[ \frac{dy}{dx} = u'v + uv' \] #### Step 1.1: Find \( u' \) and \( v' \) 1. **Differentiate \( u = 2x^2 \)**: \[ u' = \frac{d}{dx}(2x^2) = 4x \] 2. **Differentiate \( v = 2x - 1 \)**: \[ v' = \frac{d}{dx}(2x - 1) = 2 \] #### Step 1.2: Apply the Product Rule Now we can apply the product rule: \[ \frac{dy}{dx} = u'v + uv' \] Substituting \( u \), \( u' \), \( v \), and \( v' \): \[ \frac{dy}{dx} = (4x)(2x - 1) + (2x^2)(2) \] ### Step 2: Simplify the Derivative Now we will simplify the expression: 1. **First term**: \( 4x(2x - 1) = 8x^2 - 4x \) 2. **Second term**: \( 2x^2(2) = 4x^2 \) Combining these: \[ \frac{dy}{dx} = (8x^2 - 4x) + 4x^2 = 12x^2 - 4x \] ### Step 3: Evaluate the Derivative at \( x = -1 \) Now we need to find the value of the derivative at \( x = -1 \): \[ \frac{dy}{dx} \bigg|_{x = -1} = 12(-1)^2 - 4(-1) \] Calculating this: 1. \( 12(-1)^2 = 12 \) 2. \( -4(-1) = 4 \) So: \[ \frac{dy}{dx} \bigg|_{x = -1} = 12 + 4 = 16 \] ### Conclusion The derivative of the function \( y = 2x^2(2x - 1) \) at the point \( x = -1 \) is \( 16 \). ### Answer The correct option is **B. 16**. ### Explanation of Other Options - **Option A: 18** - This value does not match our calculated derivative. It may arise from a miscalculation or misunderstanding of the product rule. - **Option C: -4** - This is incorrect as it does not reflect the positive value we derived from evaluating the derivative. - **Option D: -6** - This option is also incorrect and does not correspond to any step in our calculations. ### Revision Summary - Use the product rule for derivatives when dealing with products of functions. - Differentiate each function separately before applying the product rule. - Substitute the value of \( x \) into the derivative to find the slope at that point. - Always double-check calculations to avoid common pitfalls in differentiation.