NCERT Class 12 · Ch 5 Exercise 5.5 · Q13
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.5 Question 13
Find dy/dx if yˣ=xʸ.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(- \frac{y \left(x y^{x} \log{\left(y \right)} - x^{y} y\right)}{x \left(x y^{x} - x^{y} y \log{\left(x \right)}\right)}\) where the stated powers and derivative are defined.
- Use logarithmic/chain differentiation of each variable power. Work with x,y>0 for the real variable powers.
- Differentiating with respect to x gives \(y^{x} \log{\left(y \right)} - \frac{x^{y} y}{x}\) + \(\frac{x y^{x}}{y} - x^{y} \log{\left(x \right)}\) y′=0.
- Collect y′ terms and divide by \(\frac{x y^{x}}{y} - x^{y} \log{\left(x \right)}\) when nonzero: y′=\(- \frac{y \left(x y^{x} \log{\left(y \right)} - x^{y} y\right)}{x \left(x y^{x} - x^{y} y \log{\left(x \right)}\right)}\).
Master Chapter 5: Continuity And Differentiability
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Continuity And Differentiability.
All Questions in Continuity And Differentiability — Exercise 5.5
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?