Final answer:
To find the derivative of y = ((3x + 1)³) / ((2x + 1)²), the quotient rule is applied, requiring us to first find the derivatives of the numerator and the denominator, then using the derived expressions to calculate the derivative dy/dx.
Step-by-step explanation:
To differentiate the function y = ((3x + 1)³) / ((2x + 1)²), we need to apply the quotient rule, which states that the derivative of a quotient is given by:
(d/dx)[u/v] = (v(u') - u(v')) / v²
Here, u = (3x + 1)³ and v = (2x + 1)². First, compute the derivatives u' and v':
- u' = d/dx[(3x + 1)³] = 3 * 3 * (3x + 1)² = 9(3x + 1)²
- v' = d/dx[(2x + 1)²] = 2 * 2 * (2x + 1) = 4(2x + 1)
Substitute u, u', v, and v' into the quotient rule formula to find the derivative dy/dx:
dy/dx = ((2x + 1)² * (9(3x + 1)²) - (3x + 1)³ * (4(2x + 1))) / (2x + 1)⁴
Simplify the expression and solve for dy/dx to get the final form of the derivative.