Derivative as Rate of Change

Using differentials, the approximate value of ${\left(82\right)}^{\frac{1}{4}}$ upto $3$ places of decimal would be

The approximate value of ${\left(401\right)}^{\frac{1}{2}}$, using differentials, up to $3$ places of decimals, is given by

The radius of an air bubble is increasing at the rate $1\frac{\mathrm{cm}}{s}$. At what rate is its volume increasing when the radius is $2\mathrm{cm}$ ?

If the radius of a sphere is measured as $7 \mathrm{m}$ with an error of $2 \mathrm{cm},$ then the approximate error in calculating the volume is

A $10 \mathrm{m}$ long ladder is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at a rate $3 \mathrm{cm} {\mathrm{s}}^{–1}.$ How fast is the top of the ladder sliding down along the wall, when the foot of the ladder is $8 \mathrm{m}$ away from the wall?

The approximate change in the volume of a cube of side $x$ units caused by increasing the side by $1\%$ is

The circumference of a circle is increasing with time at the rate of $20\pi {\mathrm{cms}}^{-1}$. At what rate is the area of the circle increasing when the radius is $10 \mathrm{cm} ?$

If the rate of increase of $x^3-2x^2+3x+8$ is twice the rate of increase of $x$, then values of $x$ are

If $y=5x^2+6x+6,x=2$ and $\Delta x=0.001$, then the values of $\Delta y$ and $dy$ respectively are:

If $s=60t-5t^2$ denotes the distance covered by a particle in time $\text{'}t\text{'},$ then the distance it covers before coming to rest is _____ units.

Water is running into an underground right circular conical reservoir, which is $10 \mathrm{m}$ deep and radius of its base is $5 \mathrm{m}.$ If the rate of change in the volume of water in the reservoir is $\frac{3}{2}\frac{\pi {\mathrm{m}}^3}{\min }$, then the rate (in $\frac{\mathrm{m}}{\min }$) at which water rises in it, when the water level is $4 \mathrm{m}$, is

There are two points moving along the $y-$axis which are given by $y=20+6t, y=13+t^2.$ Find velocity at which they are approaching  each other at the time of encounter (in $\mathrm{cm}/\sec$)

The approximate value of $5^{2.01}$ is $\dots \dots \dots$, where, $\left({\log }_{e}5=1.6095\right)$.

If the rate of change of area of rhombus with respect to it's side is equal to the side of rhombus, then the angles of rhombus are$\dots \dots$

The displacement is $\text{'}s\text{'}$ of the particle at time $\text{'}t\text{'}$ is given by $s=t^3-4t^2-52$. Find its velocity and acceleration at time $t=2s$.

The surface area of a spherical balloon is increasing at the rate of $2 {\mathrm{cm}}^2/\sec$. If the volume of the balloon is increasing at $a {\mathrm{cm}}^3/\sec$, when the radius of the balloon is $6 \mathrm{cm}$, then $a$ is