Category Continuity and Differentiability

22. If (y=e^{a{cos}^{-1}{x}}), (-1le xle1), show that (left(1-x^2right)frac{d^2y}{dx^2}-xfrac{dy}{dx}-a^2y=0). Continuity and Differentiability Miscellaneous Exercise Previous Next Class 12 Continuity and Differentiability Miscellaneous Exercise Differentiate w.r.t. x the function in Exercises 1 to 11. 1. (left(3x^2-9x+5right)^9). 2. ({sin}^3{x}+{cos}^6{x}). 3. (left(5xright)^{3cos{2}x}). 4. ({sin}^{-1}{left(xsqrt…

21. If (y=left|begin{matrix}fleft(xright)&gleft(xright)&hleft(xright)\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|). Continuity and Differentiability Miscellaneous Exercise Previous Next Class 12 Continuity and Differentiability Miscellaneous Exercise Differentiate w.r.t. x the function in Exercises 1 to 11. 1. (left(3x^2-9x+5right)^9). 2. ({sin}^3{x}+{cos}^6{x}). 3. (left(5xright)^{3cos{2}x}). 4. ({sin}^{-1}{left(xsqrt xright)}),(0le xle1).…

20. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer. Continuity and Differentiability Miscellaneous Exercise Previous Next Class 12 Continuity and Differentiability Miscellaneous Exercise Differentiate w.r.t. x the function in…

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. Continuity and Differentiability Miscellaneous Exercise Previous Next Class 12 Continuity and Differentiability Miscellaneous Exercise Differentiate w.r.t. x the function in Exercises 1 to 11. 1.…

18. If (fleft(xright)=left|xright|^3), show that f”(x) exists for all real x and find it. Continuity and Differentiability Miscellaneous Exercise Previous Next Class 12 Continuity and Differentiability Miscellaneous Exercise Differentiate w.r.t. x the function in Exercises 1 to 11. 1. (left(3x^2-9x+5right)^9).…

17. If (x=aleft(cos{t}+tsin{t}right) ) and (y=aleft(sin{t}-tcos{t}right)), find (frac{d^2y}{dx^2}). Continuity and Differentiability Miscellaneous Exercise Previous Next Class 12 Continuity and Differentiability Miscellaneous Exercise Differentiate w.r.t. x the function in Exercises 1 to 11. 1. (left(3x^2-9x+5right)^9). 2. ({sin}^3{x}+{cos}^6{x}). 3. (left(5xright)^{3cos{2}x}). 4. ({sin}^{-1}{left(xsqrt…

16. If (cos{y}=xcos{left(a+yright)} ) with (cos{a}neqpm1), prove that (frac{dy}{dx}=frac{{cos}^2{left(a+yright)}}{sin{a}}). Continuity and Differentiability Miscellaneous Exercise Previous Next Class 12 Continuity and Differentiability Miscellaneous Exercise Differentiate w.r.t. x the function in Exercises 1 to 11. 1. (left(3x^2-9x+5right)^9). 2. ({sin}^3{x}+{cos}^6{x}). 3. (left(5xright)^{3cos{2}x}). 4.…

15. If (left(x-aright)^2+left(y-bright)^2=c^2), for some c>0, prove that (frac{left[1+left(frac{dy}{dx}right)^2right]^frac{3}{2}}{frac{d^2y}{dx^2}}) is a constant independent of a and b. Continuity and Differentiability Miscellaneous Exercise Previous Next Class 12 Continuity and Differentiability Miscellaneous Exercise Differentiate w.r.t. x the function in Exercises 1 to…

14. If (xsqrt{1+y}+ysqrt{1+x}=0), for (-1<x<1), prove that (frac{dy}{dx}=-frac{1}{left(1+xright)^2}). Continuity and Differentiability Miscellaneous Exercise Previous Next Class 12 Continuity and Differentiability Miscellaneous Exercise Differentiate w.r.t. x the function in Exercises 1 to 11. 1. (left(3x^2-9x+5right)^9). 2. ({sin}^3{x}+{cos}^6{x}). 3. (left(5xright)^{3cos{2}x}). 4. ({sin}^{-1}{left(xsqrt…

13. Find (frac{dy}{dx}), If (y={sin}^{-1}{x}+{sin}^{-1}{sqrt{1-x^2}}),(-1le xle1). Continuity and Differentiability Miscellaneous Exercise Previous Next Class 12 Continuity and Differentiability Miscellaneous Exercise Differentiate w.r.t. x the function in Exercises 1 to 11. 1. (left(3x^2-9x+5right)^9). 2. ({sin}^3{x}+{cos}^6{x}). 3. (left(5xright)^{3cos{2}x}). 4. ({sin}^{-1}{left(xsqrt xright)}),(0le xle1).…