Let g(x)=e3x(2x2+3x)5(x2−1)1/3. A student wishes to investigate the rate of change of g with respect to x.
Respond to the prompts below, providing clear mathematical reasoning.
(a) Describe—without performing any differentiation—the algebraic steps necessary to rewrite g(x) as a sum and difference of simpler terms that are easier to differentiate. Include a brief explanation of how each step reduces the complexity of the original expression.
(b) Explain why differentiating lng(x) can be computationally more efficient than differentiating g(x) directly for this function.
(c) Identify all real values of x for which each transformation you described in part (a) is valid, and justify your conclusions.