Posted by ash on Saturday, April 30, 2011 at 9:40pm.
The product rule is:
d(uv)/dx = udv/dx+vdu/dx ...(1)
Set u=v=(3x^2-8)
dv/dx=du/dx=6x
Substitute in (1) and simplify to get final answer.
Post answer for a check if you wish.
I agree with Paul's sentiments. For some aoesrn, text books ( and more critically lecturers) often teach multiple subset rules rather than one superset rule. In my opinion it is best to state and prove a superset theorem, and then demonstrate how each of the subset rules is an application of the initial theorem. This allows students (who are so inclined) to really understand the maths, rather than remembering rules. This builds flexible and robust knowledge, while wrote learning is brittle outside of the confines of the recognised examples. There is another subtle problem with teaching multiple subset rules. It implies they are necessary, i.e. it implies the student has missed something that distinguishes these cases from the more general case. That mightn't be concerning for someone who knows they aren't necessary. But for a student learning this for the first time he must think that they are necessary. This increases the burden on memory, and for all but the most diligent students (who will seek out the unifying principle independently) it gives them a sense that they are missing something diagnostic. This means they will lack confidence in recognition of the problem type at exam time. Unfortunately the problem is rife. During my finance days at university we were told to remember at least 5 different formulae for annuity valuations, when the sum of a geometric progression was the only tool required. As another example I have often wondered why y = e^(f(x)) dy/dx = f'(x)e^(f(x)) is taught in preference to the more general: y = a^(f(x)) dy/dx = f'(x)a^(f(x))logbase e (a) {my apologies for notation but I am no html expert}. As Paul notes, both cases should be shown to have a connection with the chain rule. My hope is simple: that writers of text books prove the superset rule first, and then when writing the subset rules, draw attention to how they are no more than specific cases of that superset rule.