To find the derivative of the function \( f(x) = (2 + 3x)(1 - x) \) with respect to \( x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by: \[ (uv)' = u'v + uv' \] ### Step 1: Identify the Functions In our case, we can identify: - \( u(x) = 2 + 3x \) - \( v(x) = 1 - x \) ### Step 2: Find the Derivatives of \( u \) and \( v \) Now, we need to find the derivatives of \( u \) and \( v \): 1. **Derivative of \( u(x) \)**: \[ u'(x) = \frac{d}{dx}(2 + 3x) = 0 + 3 = 3 \] 2. **Derivative of \( v(x) \)**: \[ v'(x) = \frac{d}{dx}(1 - x) = 0 - 1 = -1 \] ### Step 3: Apply the Product Rule Now we can apply the product rule: \[ f'(x) = u'v + uv' \] Substituting the values we found: \[ f'(x) = (3)(1 - x) + (2 + 3x)(-1) \] ### Step 4: Simplify the Expression Now we will simplify the expression step by step: 1. Calculate \( 3(1 - x) \): \[ 3(1 - x) = 3 - 3x \] 2. Calculate \( (2 + 3x)(-1) \): \[ (2 + 3x)(-1) = -2 - 3x \] 3. Combine the results: \[ f'(x) = (3 - 3x) + (-2 - 3x) \] \[ f'(x) = 3 - 3x - 2 - 3x \] \[ f'(x) = 1 - 6x \] ### Final Answer Thus, the derivative of \( (2 + 3x)(1 - x) \) with respect to \( x \) is: \[ \boxed{1 - 6x} \] ### Explanation of Other Options - **Option A: 6** - This option suggests a constant derivative, which is incorrect because the function is linear in nature and its slope changes with \( x \). - **Option B: -3** - This option is also incorrect as it does not account for the variable \( x \) and suggests a constant rate of change, which is not the case here. - **Option D: 6x - 1** - This option is incorrect because it does not match the derived expression and suggests a different linear relationship. ### Revision Summary - Use the product rule for derivatives when dealing with products of functions. - Identify each function and find their derivatives separately. - Combine the results carefully, ensuring to simplify correctly. - Always check your final answer against the options provided to ensure accuracy.