Class 12 Continuity And Differentiability · Long Answer3 Marksimplicit_differentiation
Class 12 Maths · Continuity And Differentiability · Long Answer #132
If x/(x−y)=ln[a/(x−y)], where a/(x−y)>0, x−y≠0 and y≠0, prove y′=2−x/y.
✅ Step-by-Step Solution
Final Answer: 2−x/y
- Let d=x−y. Differentiate the left side by the quotient rule: [d−x(1−y′)]/d²=(xy′−y)/d².
- The derivative of ln(a/d) is −(1−y′)/d=(y′−1)/d, since a is constant.
- Equating and multiplying by d² gives xy′−y=(y′−1)(x−y). Cancelling xy′ and rearranging gives yy′=2y−x.
- Divide by y≠0: y′=2−x/y.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let d=x−y. Differentiate the left side by the quotient rule: [d−x(1−y′)]/d²=(xy′−y)/d².
- [1 mark] The derivative of ln(a/d) is −(1−y′)/d=(y′−1)/d, since a is constant.
- [1 mark] Equating and multiplying by d² gives xy′−y=(y′−1)(x−y). Cancelling xy′ and rearranging gives yy′=2y−x.
- [0.5 marks] Divide by y≠0: y′=2−x/y.
Master Chapter 5: Continuity And Differentiability
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Continuity And Differentiability.
← Previous Question
If xy=e^(x−y) on a differentiable real branch, show…
Next Question →Differentiate y=\(x^{x \cos{\left(x \right)}} + \log{\left(x…
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?