If $\sqrt{\frac{y}{x}}+\sqrt{\frac{x}{y}}=2$,then $\frac{d y}{d x}$ is equal to

$2 \tanh^{-1} \frac{1}{2}$ is equal to

The equation of the parabola with the focus $(3,0)$ and the directrix $x+3=0$ is:

If $\int \frac{7x^8+8x^7}{(1+x+x^8)^2} dx = f(x) + c$,then $f(x)$ is equal to

In the following figure,a body of mass $m$ is tied at one end of a light string and this string is wrapped around a solid cylinder of mass $M$ and radius $R$. At the moment $t = 0$,the system starts moving. If the friction is negligible,the angular velocity at time $t$ would be

$A$ certain vector in the $xy$-plane has an $x$-component of $4 ~m$ and a $y$-component of $10 ~m$. It is then rotated in the $xy$-plane so that its $x$-component is doubled. Then its new $y$-component is (approximately) (in $~m$)