text: # 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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explanation: The document contains a detailed solution to a math problem. It provides a step-by-step process to differentiate \( \tan^{-1}\left(\frac{1+2x}{1-2x}\right) \) with respect to \( \sqrt{1+4x^2} \). The solution includes substitutions, trigonometric identities, and the final answer.
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distilabel_metadata.content: # 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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  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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distilabel_metadata.raw_output_q_a_detector_0: The document contains a detailed solution to a math problem. It provides a step-by-step process to differentiate \( \tan^{-1}\left(\frac{1+2x}{1-2x}\right) \) with respect to \( \sqrt{1+4x^2} \). The solution includes substitutions, trigonometric identities, and the final answer.
  
  Label: question-and-answer