Implicitly Defined Functions Practice Test

A particle moves so its coordinates satisfy $x(t)y(t)+\delta x(t)=t$, where $\delta$ is a constant drift parameter. The position vector is $\vec r(t)=\langle x(t),y(t)\rangle$, and the constraint models motion along a time-dependent track. You want the vertical velocity component $y'(t)$ without solving for $y$. Assume $x(t)\neq 0$. Using the information provided, differentiate implicitly with respect to $t$ and solve for $y'(t)$.

A particle’s position vector is $\vec r(t)=\langle x(t),y(t)\rangle$ and is constrained by $x^2+y^2=\alpha t^2$, where $\alpha>0$ is constant. Let $\vec v(t)=\langle x'(t),y'(t)\rangle$ be velocity. Differentiate the constraint to relate $x,y$ and components of $\vec v$. Using the information provided, the motion stays on an expanding circle whose radius depends on $t$ and $\alpha$. Assume $x(t)$ and $y(t)$ are differentiable for $t>0$.​