To differentiate the function \((2x+5)^{2} (x-4)\) with respect to \(x\), we will use the **product rule** and the **chain rule**. Let's break this down step-by-step. ### Step 1: Identify the components of the function The function can be seen as a product of two functions: - Let \(u = (2x + 5)^{2}\) - Let \(v = (x - 4)\) ### Step 2: Apply the product rule The product rule states that if you have two functions \(u\) and \(v\), the derivative of their product is given by: \[ \frac{d}{dx}(uv) = u'v + uv' \] where \(u'\) is the derivative of \(u\) and \(v'\) is the derivative of \(v\). ### Step 3: Differentiate \(u\) and \(v\) 1. **Differentiate \(u = (2x + 5)^{2}\)** using the chain rule: - The chain rule states that if you have a composite function \(f(g(x))\), then the derivative is \(f'(g(x)) \cdot g'(x)\). - Here, let \(g(x) = 2x + 5\) and \(f(g) = g^{2}\). - The derivative \(f'(g) = 2g\) and \(g'(x) = 2\). - Therefore, \[ u' = 2(2x + 5) \cdot 2 = 4(2x + 5) \] 2. **Differentiate \(v = (x - 4)\)**: - The derivative of \(v\) is simply: \[ v' = 1 \] ### Step 4: Substitute into the product rule formula Now we can substitute \(u\), \(u'\), \(v\), and \(v'\) into the product rule: \[ \frac{d}{dx}((2x + 5)^{2}(x - 4)) = u'v + uv' \] Substituting the values we found: \[ = 4(2x + 5)(x - 4) + (2x + 5)^{2}(1) \] ### Step 5: Simplify the expression Now we simplify the expression: 1. The first term is: \[ 4(2x + 5)(x - 4) \] Expanding this: \[ = 4(2x^2 - 8x + 5x - 20) = 4(2x^2 - 3x - 20) = 8x^2 - 12x - 80 \] 2. The second term is: \[ (2x + 5)^{2} \] Expanding this: \[ = 4x^2 + 20x + 25 \] Now, combine both terms: \[ 8x^2 - 12x - 80 + 4x^2 + 20x + 25 = (8x^2 + 4x^2) + (-12x + 20x) + (-80 + 25) \] This simplifies to: \[ 12x^2 + 8x - 55 \] ### Step 6: Final answer The derivative of \((2x + 5)^{2}(x - 4)\) with respect to \(x\) is: \[ 12x^2 + 8x - 55 \] ### Evaluating the Options Now, let's evaluate the provided options: - **A. \(4(2x + 5)(x - 4)\)**: This is not the complete derivative; it only represents the first part of the product rule. - **B. \(4(2x + 5)(4x - 3)\)**: This is incorrect as it does not match our derived expression. - **C. \((2x + 5)(2x - 13)\)**: This is also incorrect; it does not match our derived expression. - **D. \((2x + 5)(6x - 11)\)**: This is incorrect as well; it does not match our derived expression. ### Conclusion None of the options provided are correct based on our calculations. The correct derivative is \(12x^2 + 8x - 55\). ### Revision Summary - Use the product rule for differentiating products of functions. - Apply the chain rule for differentiating composite functions. - Always simplify your final expression to ensure accuracy. - Check your work against provided options to confirm correctness.