If $x = \frac{1 + t}{t^3}$ and $y = \frac{3}{2t^2} + \frac{2}{t}$,then $x \left( \frac{dy}{dx} \right)^3 - \frac{dy}{dx}$ is equal to (where $t$ is a real parameter).

If $y=t^2+t^3$ and $x=t-t^4$,then $\frac{d^2 y}{d x^2}$ at $t=1$ is

At any two points of the curve represented parametrically by $x = a(2 \cos t - \cos 2t)$ and $y = a(2 \sin t - \sin 2t)$,the tangents are parallel to the $x$-axis. The values of the parameter $t$ corresponding to these points differ from each other by:

The derivative of $\sin x$ with respect to $\log x$ is

If $f^{\prime}(x)=\sin (\log x)$ and $y=f\left(\frac{2 x+3}{3-2 x}\right)$,then $\frac{d y}{d x}$ at $x=1$ is

If $x=a\left(t-\frac{1}{t}\right)$ and $y=b\left(t+\frac{1}{t}\right)$,then $\frac{dy}{dx}=$