I think I fully understand this problem.
y=2x(x^2+1)^1/2
Find y’
It is a combination of the product rule, the chain rule and U-substitution as I see it.
First the product rule. Two terms F,G.
F=2x
G=(x^2+1)^1/2
Therefore f(g)’ = f’g+g’f (the product rule)
But what is g’?
g’ seems to be a combination of the chain rule and u-substitution.
u = x^2+1 is a function all of its own, something is squared and then one is added.
Then the product of u is operated on by another separate function that takes the square root of the output of the u function.
The U substitution rule, as I see it is almost another way of looking at the chain rule???
I think a few moments ago I fully understood the meaning of the chain rule. I will try to sum it up in this analogy:
The chain rule differentiates multiple related functions.
For example:
Yolonda buys 3 times as many potatoes as Xander. And Xander buys 2 times as many potatoes as Ulysses. How many more potatoes does Yolonda have than Ulysses?
dY/dX = (dY/dU) (dU/dX)
This seems to be explaining the commutative property of multiplication? ( when all of the terms are functions instead of numbers )
But you can’t seem to use the chain rule without a new variable “u”
So we replace the one function “G” with two functions, “U” and (u)^1/2
Both are easy to differentiate.
The “u substitution rule” is d/dx [u^n] = n(u)^n-1 (du)
This to me seems to be directly relating The chain rule, the product rule, u-substitution, and the power rule inside the u substitution rule itself.
This implies to me that any complex function can be factored down into smaller composite functions, and then handled as multiple smaller units. Although it seems as if in doing that the order of operations of the original function seems to be lost, but perhaps I have just not fully understood that part of the rules of differentiation.