Differentiation

Step 1: Recognize the function structure.
The functions

%5Cphi_n(x)

are defined recursively:

%5Cphi_1(x)%20%3D%20e%5E%7B%5Csin%20x%7D%2C%20%5Cquad%20%5Cphi_2(x)%20%3D%20e%5E%7B%5Cphi_1(x)%7D%2C%20%5Cquad%20%5Cphi_3(x)%20%3D%20e%5E%7B%5Cphi_2(x)%7D%2C%20%5Cquad%20%5Cldots

For any

n

%5Cphi_n(x)%20%3D%20e%5E%7B%5Cphi_%7Bn-1%7D(x)%7D

.
Step 2: Apply the chain rule of differentiation.
To differentiate

%5Cphi_n(x)

, we use the chain rule. For example, to differentiate

%5Cphi_2(x)%20%3D%20e%5E%7B%5Cphi_1(x)%7D

, we get:

%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft(%20%5Cphi_2(x)%20%5Cright)%20%3D%20e%5E%7B%5Cphi_1(x)%7D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft(%20%5Cphi_1(x)%20%5Cright)%20%3D%20%5Cphi_2(x)%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft(%20e%5E%7B%5Csin%20x%7D%20%5Cright).

Step 3: Differentiate $%5Cphi_1(x)$ .
We now differentiate

%5Cphi_1(x)%20%3D%20e%5E%7B%5Csin%20x%7D

. Using the chain rule again, we get:

%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft(%20e%5E%7B%5Csin%20x%7D%20%5Cright)%20%3D%20e%5E%7B%5Csin%20x%7D%20%5Ccdot%20%5Ccos%20x.

Thus,

%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft(%20%5Cphi_1(x)%20%5Cright)%20%3D%20%5Cphi_1(x)%20%5Ccdot%20%5Ccos%20x

.
Step 4: Generalize the differentiation for $%5Cphi_n(x)$ .
Now, applying the chain rule recursively for

%5Cphi_n(x)

, we get:

%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft%5C%7B%20%5Cphi_n(x)%20%5Cright%5C%7D%20%3D%20%5Cphi_n(x)%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft%5C%7B%20%5Cphi_%7Bn-1%7D(x)%20%5Cright%5C%7D%20%5Ccdot%20%5Cldots%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft%5C%7B%20%5Cphi_1(x)%20%5Cright%5C%7D%20%5Ccdot%20%5Ccos%20x.

Thus, the derivative involves the product of all the terms up to

%5Cphi_n(x)

and is multiplied by

%5Ccos%20x

.
Step 5:
Conclusion and simplification.

Therefore, the correct answer is the expression:

%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft%5C%7B%20%5Cphi_n(x)%20%5Cright%5C%7D%20%3D%20%5Cphi_n(x)%20%5Ccdot%20%5Cphi_%7Bn-1%7D(x)%20%5Ccdot%20%5Cldots%20%5Ccdot%20%5Cphi_1(x)%20%5Ccdot%20%5Ccos%20x.

This matches option (B).

About Differentiation - TRIPURA-JEE

Differentiation is a vital chapter for TRIPURA-JEE aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Differentiation PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Differentiation carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

About Differentiation - TRIPURA-JEE

Frequently Asked Questions

Why focus on Differentiation PYQs?

How to best use this analysis?

Differentiation

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

About Differentiation - TRIPURA-JEE

Frequently Asked Questions

Why focus on Differentiation PYQs?

How to best use this analysis?

Chapter Questions
1 MCQs