Let's solve the problem step by step: **Given function:** \( y = (5x + 1)^3 (3x + 4)^{-2} \) --- ### **Step 1: Identify the rule to use** This function is a product of two functions, so we use the **product rule** and **chain rule**. Let \( u = (5x + 1)^3 \) \( v = (3x + 4)^{-2} \) The product rule states: \( \frac{d}{dx}(uv) = u'v + uv' \) --- ### **Step 2: Differentiate each part and apply the product rule** First, find \( u' \) and \( v' \) using the chain rule: - \( u = (5x + 1)^3 \implies u' = 3(5x + 1)^2 \cdot 5 = 15(5x + 1)^2 \) - \( v = (3x + 4)^{-2} \implies v' = -2(3x + 4)^{-3} \cdot 3 = -6(3x + 4)^{-3} \) Using the product rule: \[ \frac{dy}{dx} = u'v + uv' \] \[ = [15(5x + 1)^2] (3x + 4)^{-2} + (5x + 1)^3 [-6(3x + 4)^{-3}] \] Now, factor and combine into a single fraction: \[ = \frac{15(5x + 1)^2}{(3x + 4)^2} - \frac{6(5x + 1)^3}{(3x + 4)^3} \] \[ = \frac{15(5x + 1)^2 (3x + 4) - 6(5x + 1)^3}{(3x + 4)^3} \] Expand the numerator: \[ 15(5x + 1)^2 (3x + 4) - 6(5x + 1)^3 \] Let’s factor \((5x + 1)^2\) out: \[ = (5x + 1)^2 [15(3x + 4) - 6(5x + 1)] \] \[ = (5x + 1)^2 [45x + 60 - 30x - 6] \] \[ = (5x + 1)^2 [15x + 54] \] So, the simplified answer is: \[ \boxed{ \frac{3(5x + 1)^2 (5x + 18)}{(3x + 4)^3} } \] --- ### **Final Answer:** \[ \boxed{ \frac{3(5x + 1)^2 (5x + 18)}{(3x + 4)^3} } \] **Explanation:** Step 1 used the product and chain rules to find the derivatives of each part. Step 2 combined the results into a single fraction and simplified to match the form in the image.