Given $x = \dfrac{3}{2}y^3$, find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $y$.
$\dfrac{2}{9y^2}$
Given $x = \dfrac{1}{2y}$, find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $y$.
$-2y^2$
Given $x = \dfrac{y + 1}{y}$, find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $y$.
$-y^2$
Given $x = \dfrac{(y+1)(y-3)}{2y^2}$, find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $y$.
$\dfrac{y^3}{y + 3}$
The radius, $r$, of a circle is increasing at a constant rate of $3$. Find the rate at which the area of the circle is increasing when the radius is $2$.
$12\pi$
The side length, $x$, of a cube is increasing at a constant rate of $1.5$. Find the rate at which the volume of the cube is increasing when the side length is $4$.
$72$
The surface area, $S = 4\pi r^2$, of a sphere is increasing at a constant rate of $16$. Find the rate at which the radius of the sphere is increasing when the side length is $\dfrac{5}{\pi}$.
$\dfrac{2}{5}$
The radius, $r$, of a sphere is increasing at a constant rate of $2.5$. Find the rate at which the volume, $V = \dfrac{4}{3}\pi r^3$, of the sphere is increasing when the radius is $4$.
$160\pi$
The volume, $V$, of a cube is increasing at a constant rate of $6$. Find the rate at which the surface area of the cube is increasing when the side length is $8$.
$3$
The area, $A$, of a circle is increasing at a constant rate of $6$. Find the rate at which the radius of the circle is increasing when the area is $36\pi$.
$\dfrac{1}{2\pi}$
The volume, $V$, of a pile of dirt of height $h$ is given by $$V = -3 + \sqrt{h^4+9}$$ The volume is increasing at a rate of $6$. Find the rate at which the height of the pile is increasing when the height is $2$.
$\dfrac{75}{8}$
The volume, $V$, of a pile of dirt of height $h$ is given by $$V = -2 + \sqrt{h^2 + 2h}$$ The volume is increasing at a rate of $10$. Find the rate at which the height of the pile is increasing when the height is $1$.
$5\sqrt{3}$
$$y = x^2(8-x)$$ $y$ is increasing at a rate of $8$. Find the rate at which $x$ is increasing when $x = 4$.
$\dfrac{1}{2}$
$$y = \sqrt{x^3+2x^2}$$ $y$ is increasing at a rate of $5$. Find the rate at which $x$ is increasing when $x = 7$.
$\dfrac{42}{35}$
The volume of a spherical bubble is increasing at a constant rate of $3.6\pi$. Find the rate at which the radius of the bubble is increasing ten seconds after the bubble was first formed.
$0.1$
The surface area of a cube is increasing at a constant rate of $8$. Find the rate at which the volume of the cube is increasing when the surface area is $12$.
$2\sqrt{2}$