The edge of a cube is increasing at the rate of $5 \, cm/\sec$. How fast is the volume of the cube increasing when the edge is $12 \, cm$ long? (in $cm^3/\sec$)

$A$ square plate is contracting at the uniform rate $3 \,cm^2 / sec$. The rate at which the perimeter is decreasing, when the side of the square is $15 \,cm$, is

The radius of a right circular cylinder increases at the rate of $0.1 \text{ cm/min}$ and the height decreases at the rate of $0.2 \text{ cm/min}$. The rate of change of the volume of the cylinder in $\text{cm}^3\text{/min}$,when the radius is $2 \text{ cm}$ and the height is $3 \text{ cm}$,is:

The total cost $C(x)$ in Rupees,associated with the production of $x$ units of an item is given by $C(x) = 0.005x^{3} - 0.02x^{2} + 30x + 5000$. Find the marginal cost when $3$ units are produced,where marginal cost is defined as the instantaneous rate of change of total cost with respect to the number of units produced.

$A$ particle moves such that $x = 2 + 27t - t^3$. The direction of motion reverses after moving a distance of ... units.

If the rate of change of the slope of the tangent drawn to the curve $y=x^3-2x^2+3x-2$ at the point $(2,4)$ is $k$ times the rate of change of its abscissa, then $k=$