Final answer:
To validate the equation ³(d)/(dx) ln (x³+2) = (1)/(x³+2)?, the expression that should replace the question mark is 3x². This is derived by applying the chain rule to the function inside the natural logarithm.
Step-by-step explanation:
To determine what expression should replace the question mark to make the equation (d)/(dx) ln (x³+2) = (1)/(x³+2) valid, we need to apply the derivative of the natural logarithm function. The derivative of ln(u), where u is a function of x, is 1/u multiplied by the derivative of u with respect to x. Thus, if we let u = x³ + 2, then the derivative of u with respect to x is 3x². We would then have:
(d)/(dx) ln (x³+2) = 1/(x³+2) * (3x²)
So, the expression that should replace the question mark in the original equation is 3x².