Given that y = ×(2x + 1)*, show that dy = (2x + 1)" (Ax + B) dx where n, A and B are constants to be found.

1. Differentiate y = x(2x + 1) with respect to x. 2. Set the result equal to (2x + 1) (Ax + B). Let's go through the steps: 1. У = x(2x + 1) Apply the product rule: Y = (2x + 1) + x (2) 2. Simplify the derivative: 3=22+1+22 3. Combine like terms: У = 4x + 1 Now, set this equal to (2x + 1) (Ax + B): 4x + 1 = (2x + 1)(Ax + B) Expand the right side: 4x + 1 = 2Ax% + (2A + B)x + B Now, match coefficients: 1. Coefficient of x?: 2A = O implies A = 0 2. Coefficient of x: 2A + B = 4 implies B = 4 3. Constant term: B = 1 so, A = 0 and B = 4. The expression for dy/ d is: dy/dx = (2x + 1)(Ax + B) = (2x + 1)(4x + 4