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Differentiation
Class XII · Exercise 11.4 · Implicit Functions
Exercise 11.4
Level 1
Find \(\dfrac{dy}{dx}\) in each of the following (Q. 1–11):
- 1. \( xy = c^2 \)
- 2. \( y^3 - 3xy^2 = x^3 + 3x^2 y \)
- 3. \( x^{2/3} + y^{2/3} = a^{2/3} \)
- 4. \( 4x + 3y = \log(4x - 3y) \)
- 5. \( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \)
- 6. \( x^5 + y^5 = 5xy \)
- 7. \( (x + y)^2 = 2axy \)
- 8. \( (x^2 + y^2)^2 = xy \) CBSE 2009
- 9. \( \tan^{-1}(x^2 + y^2) = a \)
- 10. \( e^{x-y} = \log\!\left(\dfrac{x}{y}\right) \)
- 11. \( \sin xy + \cos(x + y) = 1 \)
- 12. If \(\sqrt{1-x^2} + \sqrt{1-y^2} = a(x-y)\), prove that \(\dfrac{dy}{dx} = \sqrt{\dfrac{1-y^2}{1-x^2}}\). NCERT Exemplar
Answers
Exercise 11.4 — Level 1
1. \(-\dfrac{y}{x}\)
2. \(\dfrac{dy}{dx} = \dfrac{(x+y)^2}{y^2 - 2xy - x^2}\)
3. \(\left(-\dfrac{y}{x}\right)^{1/3}\)
4. \(\dfrac{4(1-4x+3y)}{3(4x-3y+1)}\)
5. \(-\dfrac{b^2 x}{a^2 y}\)
6. \(\dfrac{y - x^4}{y^4 - x}\)
7. \(\dfrac{ay - x - y}{x + y - ax}\)
8. \(\dfrac{4x(x^2+y^2)-y}{x - 4y(x^2+y^2)}\)
9. \(-\dfrac{x}{y}\)
10. \(\dfrac{y}{x}\cdot\dfrac{xe^{x-y}-1}{ye^{x-y}-1}\)
11. \(\dfrac{\sin(x+y) - y\cos(xy)}{x\cos(xy) - \sin(x+y)}\)
Exercise 11.4
Level 2 — Prove / Show / Find
- 13. If \(y\sqrt{1-x^2} + x\sqrt{1-y^2} = 1\), prove that \(\dfrac{dy}{dx} = -\sqrt{\dfrac{1-y^2}{1-x^2}}\).
- 14. If \(xy = 1\), prove that \(\dfrac{dy}{dx} + y^2 = 0\).
- 15. If \(xy^2 = 1\), prove that \(2\dfrac{dy}{dx} + y^3 = 0\).
- 16. If \(x\sqrt{1+y} + y\sqrt{1+x} = 0\), prove that \((1+x)^2\dfrac{dy}{dx} + 1 = 0\). CBSE 2011
- 17. If \(\log\sqrt{x^2+y^2} = \tan^{-1}\!\left(\dfrac{y}{x}\right)\), prove that \(\dfrac{dy}{dx} = \dfrac{x+y}{x-y}\).
- 18. If \(\sec\!\left(\dfrac{x+y}{x-y}\right) = a\), prove that \(\dfrac{dy}{dx} = \dfrac{y}{x}\).
- 19. If \(\tan^{-1}\!\left(\dfrac{x^2-y^2}{x^2+y^2}\right) = a\), prove that \(\dfrac{dy}{dx} = \dfrac{x(1-\tan a)}{y(1+\tan a)}\).
- 20. If \(xy\log(x+y) = 1\), prove that \(\dfrac{dy}{dx} = -\dfrac{y(x^2y + x + y)}{x(xy^2 + x + y)}\).
- 21. If \(y = x\sin(a+y)\), prove that \(\dfrac{dy}{dx} = \dfrac{\sin^2(a+y)}{\sin(a+y) - y\cos(a+y)}\).
- 22. If \(x\sin(a+y) + \sin a\cos(a+y) = 0\), prove that \(\dfrac{dy}{dx} = \dfrac{\sin^2(a+y)}{\sin a}\). NCERT Exemplar
- 23. If \(y = x\sin y\), prove that \(\dfrac{dy}{dx} = \dfrac{\sin y}{1 - x\cos y}\).
- 24. If \(y\sqrt{x^2+1} = \log\!\left(\sqrt{x^2+1} - x\right)\), show that \((x^2+1)\dfrac{dy}{dx} + xy + 1 = 0\).
- 25. If \(\sin(xy) + \dfrac{y}{x} = x^2 - y^2\), find \(\dfrac{dy}{dx}\).
- 26. If \(\tan(x+y) + \tan(x-y) = 1\), find \(\dfrac{dy}{dx}\).
- 27. If \(e^x + e^y = e^{x+y}\), prove that \(\dfrac{dy}{dx} = -\dfrac{e^x(e^y-1)}{e^y(e^x-1)}\) or \(\dfrac{dy}{dx} + e^{y-x} = 0\). CBSE 2014
- 28. If \(\cos y = x\cos(a+y)\), with \(\cos a \neq \pm 1\), prove that \(\dfrac{dy}{dx} = \dfrac{\cos^2(a+y)}{\sin a}\). NCERT
- 29. If \(y = \{\log_{\cos x}\sin x\}\{\log_{\sin x}\cos x\}^{-1} + \sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right)\), find \(\dfrac{dy}{dx}\) at \(x = \dfrac{\pi}{4}\).
- 30. If \(\sqrt{y+x} + \sqrt{y-x} = c\), show that \(\dfrac{dy}{dx} = \dfrac{y}{x} - \sqrt{\dfrac{y^2}{x^2} - 1}\).
Answers
Selected — Level 2
25. \(\dfrac{2x^3 + y - x^2 y\cos(xy)}{x\{x^2\cos xy + 1 + 2xy\}}\)
26. \(\dfrac{\sec^2(x-y)+\sec^2(x+y)}{\sec^2(x-y)-\sec^2(x+y)}\)
29. \(8\!\left\{\dfrac{4}{\pi^2+16} - \dfrac{1}{\log 2}\right\}\)
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