Question

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

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Cristian

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50 Answers

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

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