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

Question 13. Find \frac{dy}{dx} , if y = \sin^{-1}x + \sin^{-1}{\sqrt{1 – x^2}}, 0<x<1

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

Question 15. If (x – a)^2 + (y – b)^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.

Question 16. If \cos y = x \cos (a + y) , with \cos a ne pm 1 , prove that \frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a} .

Question 17. If x = a (\cos t + t \sin t) rm{and} y = a (\sin t – t \cos t), find \frac{d^2{x}}{dx^2} .

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

Question 19. Using mathematical induction prove that \frac{d}{dx}(x^n) = nx^{n – 1} for all positive integers n.

Question 20. Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Question 21. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.

Question 22. If \begin{vmatrix} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{vmatrix} prove that \frac{dy}{dx} = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\  l & m & n \\ a & b & c \end{vmatrix}

Question 23. If y = e^{a \cos^{-1}x}, -1 le x le 1 , show that (1 – x^2) \frac{d^2{y}}{dx^2} – x \frac{dy}{dx} – a^2 y = 0