Final answer:
To find the third derivative of the function f(x) = 2x^2ln(x), one must apply differentiation rules three times. The correct third derivative is 4xln(x) + 2x^2, which is derived using the product rule, power rule, and the fact that the derivative of ln(x) is 1/x. The correct multiple-choice option is (A).
Step-by-step explanation:
To find f[3](x), the third derivative of the function f(x) = 2x2ln(x), we need to use the rules of exponents and logarithms to differentiate successively three times. Firstly, applying the product rule along with the use of the power rule and the chain rule. Next, we use the fact that the derivative of ln(x) is 1/x. So, after differentiating three times, the third derivative should be the expression that represents the rate at which the rate of change of the rate of change of f(x) is changing with respect to x.
The correct answer to the student's question after performing differentiation thrice is option A, which is 4xln(x) + 2x2.