Differentiation

Step 1: Recognize the function structure.
The functions are defined recursively: For any , .
Step 2: Apply the chain rule of differentiation.
To differentiate , we use the chain rule. For example, to differentiate , we get:
Step 3: Differentiate .
We now differentiate . Using the chain rule again, we get: Thus, .
Step 4: Generalize the differentiation for .
Now, applying the chain rule recursively for , we get: Thus, the derivative involves the product of all the terms up to and is multiplied by .
Step 5:
Conclusion and simplification.

Therefore, the correct answer is the expression: This matches option (B).