Class 12 Continuity And Differentiability · Long Answer3 Marksimplicit_differentiation
Class 12 Maths · Continuity And Differentiability · Long Answer #80
If e^x+e^y=e^(x+y), prove y′+e^(y−x)=0.
✅ Step-by-Step Solution
Final Answer: y′=−e^(y−x)
- Differentiate implicitly: e^x+e^y y′=e^(x+y)(1+y′).
- Collect: [e^y−e^(x+y)]y′=e^(x+y)−e^x.
- Use e^(x+y)=e^x+e^y to replace the two sides by −e^x y′=e^y; since e^x>0, divide safely.
- Thus y′=−e^(y−x), proving the equation.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Differentiate implicitly: e^x+e^y y′=e^(x+y)(1+y′).
- [1 mark] Collect: [e^y−e^(x+y)]y′=e^(x+y)−e^x.
- [1 mark] Use e^(x+y)=e^x+e^y to replace the two sides by −e^x y′=e^y; since e^x>0, divide safely.
- [0.5 marks] Thus y′=−e^(y−x), proving the equation.
Master Chapter 5: Continuity And Differentiability
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Continuity And Differentiability.
← Previous Question
If x sin(a+y)+sin a cos(a+y)=0, prove y′=sin²(a+y)/sin a where sin…
Next Question →For x=\(e^{\cos{\left(2 t \right)}}\), y=\(e^{\sin{\left(2 t…
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?