SUMMARY
The differentiation of the function f(x) = 2x^(2/3)(3-4x^(1/3)) requires the application of the product rule. The correct derivative is obtained by applying the product rule as f' = g'h + gh', where g = 2x^(2/3) and h = (3-4x^(1/3)). The final expression for the derivative is f'(x) = (4/3)(x^(-1/3))(3-4x^(1/3)) + (1/3)(-4)(x^(-2/3)), which simplifies to (4/x^(1/3)) - (16/3) - (4/3)(x^(-2/3)). The mistake identified was the failure to multiply the second term by 2x^(2/3) during the differentiation process.
PREREQUISITES
- Understanding of the product rule in calculus
- Familiarity with differentiation of polynomial functions
- Knowledge of exponent rules and negative exponents
- Basic algebraic manipulation skills
NEXT STEPS
- Review the product rule in calculus for differentiating products of functions
- Practice differentiating functions with fractional exponents
- Explore the implications of negative exponents in calculus
- Study examples of common differentiation mistakes and how to avoid them
USEFUL FOR
Students studying calculus, particularly those focusing on differentiation techniques, and educators seeking to clarify the product rule and its application in polynomial functions.