Rate of Change of Quantities

What is the derivative of ${\sin }^{2}x$ with respect to ${\cos }^{2}x$?

The point $P\left(x, y\right)$ is moving along the curve $y=x^2-\frac{10}{3}{x}^{\frac{3}{2}}+5x$ in such a way that the rate of change of $y$ is constant. Find the values of $x$ at the point at which the rate of change of $x$ is equal to half the rate of change of $y$.

The length $x$ of a rectangle is decreasing at the rate of $3 \mathrm{cm}/$minute and the width $y$ is increasing at the rate of $2 \mathrm{cm}/$minute. When $x=10 \mathrm{cm}$ and $y=6 \mathrm{cm}$, find the rates of change of the perimeter.

A tree is growing so that, after $t$-years its height is increasing at a rate of $\frac{18}{\sqrt{t}} \mathrm{cm}$ per year.

Assume that when $t=0$, the height is $5 \mathrm{cm}$

Find the height of the tree after $4 \mathrm{years}$ and after how many years will the height be $149 \mathrm{cm}$?

During Holi festival, a boy uses pichkaari water and fills a balloon whose radius increases at a rate of $3 {\mathrm{cms}}^{-1}$. When radius $R$ of balloon is $2\mathrm{cm}$, find the rate at which its surface area increases.

A variable triangle is inscribed in a circle of radius $R$. If the rate of change of a side is $R$ times the rate of change of the opposite angle, then that angle, is

If $\mathrm{s}={\mathrm{t}}^3-4{\mathrm{t}}^2+5$ describes the motion of a particle, then its velocity (in unit/sec) when the acceleration vanishes, is

An edge of a cube is increasing at the rate of $3 \mathrm{cm}/\sec$. Find the rate at which does the volume increase (in ${\mathrm{cm}}^3/\sec$) if the edge of the cube is $10 \mathrm{cm}$.

The rate of change of volume of sphere with respect to its radius $r$ at $r=2$ is

A circular disc of radius $3 \mathrm{cm}$ is being heated. Due to expansion, its radius increases at the rate of $0.05 \mathrm{cm}/\mathrm{s}$. Find the rate at which its area is increasing when radius is $3.2 \mathrm{cm}$.

A man $2$ metres high walks at a uniform speed of $5 \mathrm{km}/\mathrm{hr}$ away from a lamp-post $6$ metres high. Find the rate at which the length of his shadow increases.

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is ${\tan }^{-1}\left(0.5\right)$. Water is poured into it at a constant rate of $5$ cubic metre per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is $4 \mathrm{m}$.

A car starts from a point $P$ at time $t=0$ seconds and stops at point $Q$. The distance $x$, in metres, covered by it, in $t$ seconds is given by

$x=t^2\left(2-\frac{t}{3}\right)$

Find the time taken by it to reach $Q$ and also find distance between $P$ and $Q$.

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

The total cost $\mathrm{C}\left(x\right)$ in Rupees, associated with the production of $x$ units of an item is given by

$C\left(x\right)=0.005x^3-0.02x^2+30x+5000$

Find the marginal cost when $3$ units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

The length $x$ of a rectangle is decreasing at the rate of $3 \mathrm{cm}/$minute and the width $y$ is increasing at the rate of $2 \mathrm{cm}/$minute. When $x=10 \mathrm{cm}$ and $y=6 \mathrm{cm}$, find the rates of change of  the area of the rectangle.

A stone is dropped into a quiet lake and waves move in circles at a speed of $4 \mathrm{cm}$ per second. At the instant, when the radius of the circular wave is $10 \mathrm{cm}$, how fast is the enclosed area increasing?

The volume of a cube is increasing at a rate of $9$ cubic centimetres per second. How fast is the surface area increasing when the length of an edge is $10$ centimetres ?