Continuity and ...

lf $$y=\sqrt{2x-x^{2}}$$ then $$y^{3}.y''=$$

$${A} :$$ If $$x = ct, y = \dfrac{c}{t}$$, then at $$t=1,  \dfrac{dy}{dx} =$$
$$B:$$ If $$x=3\cos\theta -\cos^{3}\theta, y=3\sin\theta-\sin^{3}\theta$$, then at $$\theta = \dfrac{\pi}{3}, \dfrac{dy}{dx} = $$
$$C:$$ If $$x = a\left(t+ \dfrac{1}{t}\right), y = a\left(t-\dfrac{1}{t}\right)$$, then at $$t =2, \dfrac{dy}{dx} = $$
$$D:$$ Derivative of $$\log(\sec x)$$ with respect to $$\tan x$$ at $$x = \dfrac{\pi}{4}$$ is$$\\$$
Arrangement of the above values in the increasing order is

Let $$f(x)=\sin x,g(x)=x^{2},\ h(x)=\log x$$,If $$F(x)=h\{f(g(x))\}$$, then $$\displaystyle \frac{d^{2}F}{dx^{2}}=$$

If $$t\left( 1+{ x }^{ 2 } \right) =x$$ and $${ x }^{ 2 }+{ t }^{ 2 }=y,$$ then at $$x=2,$$ the value of $$\displaystyle\frac{dy}{dx}$$

If $$\mathrm{x}=\mathrm{a}\mathrm{t}^{2},\ \mathrm{y}=2\mathrm{a}\mathrm{t}$$, then $$\displaystyle \frac{\mathrm{d}^{2}\mathrm{y}}{\mathrm{d}\mathrm{x}^{2}}$$ is

 $$\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{x}}[l \mathrm{o}\mathrm{g}(\mathrm{a}\mathrm{x})^{\mathrm{x}}]$$, where $$a$$ is a constant, is equal to

 If $$r=\sqrt{x^{2}+y^{2}+z^{2}}$$ and $$x=2\sin 3t,y=2\cos 3t,z=8t$$  then $$\displaystyle \frac{dr}{dt}=$$

If $$ x=\cos\theta+\theta\sin\theta$$, $$ y=\sin\theta-\theta\cos\theta$$, then $${y}_{2}$$ is

If $$y=\sin^{-1}x\ then \ (1-x^{2})\displaystyle \frac{d^{2}y}{dx^{2}}=$$

lf $$y=\sqrt{\cos 2x}$$, then $$y\displaystyle \frac{d^{2}y}{dx^{2}}+2y^{2}=$$