Class 12 Continuity And Differentiability · Long Answer3 Marksbranch_sensitive_differentiation
Class 12 Maths · Continuity And Differentiability · Long Answer #143
For 0<|x|<1, differentiate y=arctan[(√(1+x²)+√(1−x²))/(√(1+x²)−√(1−x²))].
✅ Step-by-Step Solution
Final Answer: −x/√(1−x⁴)
- Let θ=arcsin(x²)∈(0,π/2). The sum-over-difference half-angle quotient is cot(θ/2).
- Since θ/2∈(0,π/4), the principal arctangent of cot(θ/2) is π/2−θ/2. Thus y=π/2−½arcsin(x²).
- Differentiate: y′=−½·2x/√(1−x⁴). The original denominator is zero at x=0, which is excluded.
- Therefore y′=−x/√(1−x⁴).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let θ=arcsin(x²)∈(0,π/2). The sum-over-difference half-angle quotient is cot(θ/2).
- [1 mark] Since θ/2∈(0,π/4), the principal arctangent of cot(θ/2) is π/2−θ/2. Thus y=π/2−½arcsin(x²).
- [1 mark] Differentiate: y′=−½·2x/√(1−x⁴). The original denominator is zero at x=0, which is excluded.
- [0.5 marks] Therefore y′=−x/√(1−x⁴).
Master Chapter 5: Continuity And Differentiability
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Continuity And Differentiability.
← Previous Question
Differentiate u=arctan[x/√(1−x²)] with respect to…
Next Question →Differentiate u=arctan[(√(1+x²)−1)/x] with respect to…
More Practice Questions in Continuity And Differentiability
Related Questions in Continuity And Differentiability
Class 12 Continuity And Differentiability · Mcq Single Correct
For k<0, define f(x)=(1−cos(kx))/(x sin x) for nonzero x near 0, and f(0)=1/2. Find k for…
Class 12 Continuity And Differentiability · Mcq Single Correct
Near π/2, f(x)=k cos x/(π−2x) for x≠π/2 and f(π/2)=3. Find k for continuity.
Class 12 Continuity And Differentiability · Mcq Single Correct
Find the full set of real inputs excluded from the natural domain of cot x (its poles).
Class 12 Continuity And Differentiability · Mcq Single Correct
How many real inputs are excluded from the original formula f(x)=(x−1)/[x(x²−1)]?
Class 12 Continuity And Differentiability · Mcq Single Correct
For x≠0, f(x)=(e^(3x)−e^(−5x))/x, and f(0)=k. Find k for continuity at 0.
Class 12 Continuity And Differentiability · Mcq Single Correct
Let f(x)=x/|x| for x<0 and f(x)=−1 for x≥0. What is its complete continuity set?