Derivatives of Functions in Parametric Forms

Find $\frac{dy}{dx}$ of the function expressed in parametric form $\sin x=\frac{2t}{1+t^2}, \tan y=\frac{2t}{1-t^2}$.

Find $\frac{dy}{dx}$ of the function expressed in parametric form

$x=3\cos \theta -2{\cos }^3\theta , y=3\sin \theta -2{\sin }^3\theta$

Find $\frac{dy}{dx}$ of the function expressed in parametric form $x=e^{\theta }\left(\theta +\frac{1}{\theta }\right), y=e^{-\theta }\left(\theta -\frac{1}{\theta }\right)$.

Find $\frac{dy}{dx}$ of the function expressed in parametric form

$x=t+\frac{1}{t}, y=t-\frac{1}{t}$

If $x=3\sin t-\sin 3t,y=3\cos t-\cos 3t$, if $\frac{dy}{dx}$ at $t=\frac{\pi }{3}$ is of the form of $-\frac{1}{\sqrt{m}}$, then $m=$

Find $\frac{dy}{dx}$ of the function expressed in parametric form $x=\frac{1+\log t}{t^2}, y=\frac{3+2\log t}{t}$.

If $x=a\sin 2t(1+\cos 2t)$ and $y=b\cos 2t(1-\cos 2t)$, show that ${\left(\frac{dy}{dx}\right)}_{\mathrm{at} t=\frac{\mathrm{\pi }}{4}}=\frac{b}{a}$.

If $x=e^{\cos \left(2t\right)}$ and $y=e^{\sin \left(2t\right)}$ then prove that $\frac{dy}{dx}=-\frac{y\log x}{x\log y}$.

If $f\left(x\right)=\left|\cos x\right|$ and $f^{\text{'}}\left(\frac{\pi }{4}\right)=-\frac{1}{\sqrt{k}}$, then $k=$_____.

If $x=t^2,y=t^3$, then $\frac{d^{2}y}{d{x}^2}$ is 

If $x=\sin t$ and $y=\sin pt$, Prove that $\left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+p^{2}y=0$.