The rate of change of the volume of a sphere with respect to its surface area,when its radius is $2 \text{ cm}$,is . . . . . . $\text{cm}^3 / \text{cm}^2$.

The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 3x^2 + 36x + 5$. Find the marginal revenue when $x = 5$,where marginal revenue is defined as the rate of change of total revenue with respect to the number of items sold at an instant.

$A$ ladder of $5 \ m$ long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of $3 \ m/sec$. The height of the upper end (in meters) while it is descending at the rate of $4 \ m/sec$,is

If the area of a circle increases at a uniform rate,then prove that the rate of change of its perimeter varies inversely as the radius.

The height of a right circular cylinder is decreasing while its diameter is increasing at a rate of $4 \text{ cm/s}$ so as to keep its volume unchanged. The rate of change in its lateral surface area (in $\text{cm}^2/\text{s}$) at the instant when its diameter is $8 \text{ cm}$ and height is $12 \text{ cm}$, is (in $\pi$)

If a particle moves such that the displacement $s$ is proportional to the square of the velocity $v$ acquired,then its acceleration is