# Differentiate tan−1(1+2x1−2x) with respect to √1+4x2 .

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## To differentiate tan−1(1+2x1−2x) with respect to √1+4x2, we will follow these steps:Step 1: Define the FunctionsLet:- u=tan−1(1+2x1−2x)- v=√1+4x2We need to find dudv.Step 2: Substitute for SimplicityTo simplify u, we can use a substitution. Let:2x=tan(θ)Then:x=12tan(θ)Step 3: Rewrite uNow, substituting 2x in u:u=tan−1(1+tan(θ)1−tan(θ))Using the identity tan−1(tan(a)+tan(b)1−tan(a)tan(b))=tan−1(tan(a+b)), we can set a=π4 (since tan(π4)=1):u=tan−1(tan(π4+θ))=π4+θStep 4: Rewrite vNow, for v:v=√1+4x2=√1+4(12tan(θ))2=√1+tan2(θ)=√sec2(θ)=sec(θ)Step 5: Differentiate u and vNow, we differentiate u and v with respect to θ:1. Differentiate u: dudθ=12. Differentiate v: dvdθ=sec(θ)tan(θ)Step 6: Find dudvNow, we can find dudv:dudv=du/dθdv/dθ=1sec(θ)tan(θ)Step 7: Substitute BackRecall that tan(θ)=2x and sec(θ)=√1+4x2:dudv=1√1+4x2⋅(2x)=12x√1+4x2Final AnswerThus, the derivative dudv is:dudv=12x√1+4x2---

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The text passage contains a detailed step-by-step solution to a calculus problem, specifically differentiating a function with respect to another function.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Differentiate tan−1(1+2x1−2x) with respect to √1+4x2 .

Video Solution
Text Solution
Generated By DoubtnutGPT

## To differentiate tan−1(1+2x1−2x) with respect to √1+4x2, we will follow these steps:Step 1: Define the FunctionsLet:- u=tan−1(1+2x1−2x)- v=√1+4x2We need to find dudv.Step 2: Substitute for SimplicityTo simplify u, we can use a substitution. Let:2x=tan(θ)Then:x=12tan(θ)Step 3: Rewrite uNow, substituting 2x in u:u=tan−1(1+tan(θ)1−tan(θ))Using the identity tan−1(tan(a)+tan(b)1−tan(a)tan(b))=tan−1(tan(a+b)), we can set a=π4 (since tan(π4)=1):u=tan−1(tan(π4+θ))=π4+θStep 4: Rewrite vNow, for v:v=√1+4x2=√1+4(12tan(θ))2=√1+tan2(θ)=√sec2(θ)=sec(θ)Step 5: Differentiate u and vNow, we differentiate u and v with respect to θ:1. Differentiate u: dudθ=12. Differentiate v: dvdθ=sec(θ)tan(θ)Step 6: Find dudvNow, we can find dudv:dudv=du/dθdv/dθ=1sec(θ)tan(θ)Step 7: Substitute BackRecall that tan(θ)=2x and sec(θ)=√1+4x2:dudv=1√1+4x2⋅(2x)=12x√1+4x2Final AnswerThus, the derivative dudv is:dudv=12x√1+4x2---

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NCERT solutions for CBSE and other state boards is a key requirement for students. Doubtnut helps with homework, doubts and solutions to all the questions. It has helped students get under AIR 100 in NEET & IIT JEE. Get PDF and video solutions of IIT-JEE Mains & Advanced previous year papers, NEET previous year papers, NCERT books for classes 6 to 12, CBSE, Pathfinder Publications, RD Sharma, RS Aggarwal, Manohar Ray, Cengage books for boards and competitive exams.
Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi medium and English medium for IIT JEE and NEET preparation

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The text passage contains a detailed step-by-step solution to a calculus problem, specifically differentiating a function with respect to another function.

TYPE: SINGLE_QUESTION_WITH_ANSWER