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13. Find \frac{dy}{dx}, If y={sin}^{-1}{x}+{sin}^{-1}{\sqrt{1-x^2}},-1\le\ x\le1.

Continuity and Differentiability

Miscellaneous Exercise

ncert solutions class 12 chapter 5 continuity and differentiability miscellaneous exercise question 13

Class 12

Continuity and Differentiability

Miscellaneous Exercise

Differentiate w.r.t. x the function in Exercises 1 to 11.

1. \left(3x^2-9x+5\right)^9.

2. {sin}^3{x}+{cos}^6{x}.

3. \left(5x\right)^{3cos{2}x}.

4. {sin}^{-1}{\left(x\sqrt x\right)},0\le\ x\le1.

5. \frac{{cos}^{-1}{\frac{x}{2}}}{\sqrt{2x+7}},-2<x<2.

6. {cot}^{-1}{\left[\frac{\sqrt{1+sin{x}}+\sqrt{1-sin{x}}}{\sqrt{1+sin{x}}-\sqrt{1-sin{x}}}\right]},0<x<\frac{\pi}{2}.

7. \left(log{x}\right)^{log{x}}, x>1.

8. cos{\left(acos{x}+bsin{x}\right)}, for some constant a and b.

9. \left(sin{x}-cos{x}\right)^{\left(sin{x}-cos{x}\right)}, \frac{\pi}{4}<x<\frac{3\pi}{4}.

10. x^x+x^a+a^x+a^a, for some fixed a>0 and x>0.

11. x^{x^2-3}+\left(x-3\right)^{x^2}, for x>3.

12. Find \frac{dy}{dx}, if y=12\left(1-cos{t}\right),x=10\left(t-sin{t}\right),-\frac{\pi}{2}<t<\frac{\pi}{2}.

13. Find \frac{dy}{dx}, If y={sin}^{-1}{x}+{sin}^{-1}{\sqrt{1-x^2}},-1\le\ x\le1.

14. If x\sqrt{1+y}+y\sqrt{1+x}=0, for -1<x<1, prove that \frac{dy}{dx}=-\frac{1}{\left(1+x\right)^2}.

15. If \left(x-a\right)^2+\left(y-b\right)^2=c^2, for some c>0, prove that \frac{\left[1+\left(\frac{dy}{dx}\right)^2\right]^\frac{3}{2}}{\frac{d^2y}{dx^2}} is a constant independent of a and b.

16. If cos{y}=xcos{\left(a+y\right)} with cos{a}\neq\pm1, prove that \frac{dy}{dx}=\frac{{cos}^2{\left(a+y\right)}}{sin{a}}.

17. If x=a\left(cos{t}+tsin{t}\right) and y=a\left(sin{t}-tcos{t}\right), find \frac{d^2y}{dx^2}.

18. If f\left(x\right)=\left|x\right|^3, show that f”(x) exists for all real x and find it.

19. Using the fact that \sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B} and the differentiation, obtain the sum formula for cosines.

20. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.

21. If y=\left|\begin{matrix}f\left(x\right)&g\left(x\right)&h\left(x\right)\\l&m&n\\a&b&c\\\end{matrix}\right|, prove that \frac{dy}{dx}=\left|\begin{matrix}f'(x)&g'(x)&h'(x)\\l&m&n\\a&b&c\\\end{matrix}\right|.

22. If y=e^{a{cos}^{-1}{x}}, -1\le\ x\le1, show that \left(1-x^2\right)\frac{d^2y}{dx^2}-x\frac{dy}{dx}-a^2y=0.