A point is moving along the curve $y=\frac{2}{x}+5x$ in such a way that the $x-$coordinate is increasing at a constant rate of $0.02$ units per second. Find the rate at which the $y-$coordinate is changing when $x=2$, stating whether the $y-$coordinate is increasing or decreasing.

A point moves along the curve $y=\frac{8}{7-2x}$ As it passes through the point $P$, the $x-$coordinate is increasing at a rate of $0.125$ units per second and the $y-$coordinate is increasing at a rate of $0.08$ units per second. Find the possible $x-$coordinates of $P$.

A point, $P$, travels along the curve $y=\sqrt[3]{2x^2-3}$ in such a way that at time $t$ minutes the $x-$coordinate of $P$ is increasing at a constant rate of $0.012$ units per minute. Find the rate at which the $y-$coordinate of $P$ is changing when $P$ is at the point $(1,-1)$.

A point $P(x, y)$, travels along the curve $y=x^3-5x^2+5x$ in such a way that the rate of change of $x$ is constant. Find the values of $x$ at the points where the rate of change of $y$ is double the rate of change of $x$.

A circle has radius $r$ cm and area $A {\text{cm}}^2$. The radius is increasing at a rate of $0.1 \text{cm}/\text{s}$. Find the rate of increase of $A$ when $r=4$ cm.

A sphere has radius $r$ cm and volume $V {\mathrm{cm}}^3$. The radius is increasing at a rate of $\frac{1}{2\pi }\mathrm{cm}/\mathrm{s}$. Find the rate of increase of the volume when $V=36\pi$.

A cone has radius $r$ cm and a fixed height of $30$ cm. The radius is increasing at a rate of $0.01 \mathrm{cm}/\mathrm{s}$. Find the rate of increase of the volume when $r=5$ cm.

A square has side length $x$ cm and area $A {\mathrm{cm}}^2$. The area is increasing at a constant rate of $0.03 {\mathrm{cm}}^2/\mathrm{s}$. Find the rate of increase of $x$ when $A=25 {\mathrm{cm}}^2$.

A cube has side length $x$ cm and volume $V {\mathrm{cm}}^3$. The volume is increasing at a rate of $1.5 {\mathrm{cm}}^3/\mathrm{s}$. Find the rate of increase of $x$ when $V=8 {\mathrm{cm}}^3$.