Here is what you get as an answer usually when you ask a question on Calculus or even Physics in my experience and evidently the asker was satisfied. The person answering is a professor at a college. This girl asked on line for help on what the derivative of 4cos(5x2) was. This is a chain rule problem. How would you like it answered?

Best answer as selected by question asker.
For a function f(x) = g(h(x)), express h(x) as y.
Then f(x) = g(y), f’(x) = [d {g(y)}/ dy]*(dy/dx).
Here we have to find the derivative of f(x)= 4 cos (5x2).
Let y=5x2, this gives f(x)= 4 cos y
f’(x)= [d (4 cos y)/dy]*[d(5x2)/dx]
We also know that the derivative of cos x= sin x.
=> [d (4 cos y)/dy]= 4 sin y
[d(5x2)/dx]= 5
Therefore f’(x)= [d (4 cos y)/dy]*[d(5x2)/dx]
= (4 sin y)*5
=4*sin (5x2)*5
=20 sin (5x2)
Therefore the derivative of 4 cos (5x2) is 20 sin (5x2)
Silvi says
I agree with Paul’s sentiments. For some rosaen, 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.
Dabean says
Paul I agree that simpler is betetr, so I thought about this quite a bit. In the end, I felt it was important for students to learn to apply the Chain Rule by pure reflex for those special cases that come up repeatedly after all, developing the right pavlovian reflexes is also part of learning mathematics. But there are no absolutes in pedagogy, and I certainly acknowledge your point (BTW, for the 2nd Edition currently in progress, I did reduce this slightly).
Dyessybell says
Brings back college days Calculus was my fatovire subject, but now I wonder how I actually made it through. I am now going to play around with the chain rule for a bit and see how it stacks up. Thank you for reigniting the old math spark!Charlie