Problem understanding the use of differentiation I am new to differentiation. Our physics teacher gave us this example problem:

The radius of a sphere is continuously increasing at the rate of 1 m/sec. Find the rate of change of the volume of the sphere when the radius is $\frac{1}{\sqrt{\pi}}\,\mathrm{m}$ (Say $K$).

The answer using differentiation is $4\,\mathrm{m^3/s}$.
But, as far as I understand, dx is very small change (infinitely small?) in x. So why does differentiation have so much significance in physics, if we are only dealing with infinitesimal small changes? Also, what does it mean to find the rate of change of a quantity at a particular value (K here)? What’s the use?
 A: Dealing with infinitesmally small values, in principle, allows you to is start by mathematically describing something on a very small, localized scale which often simplifies the math (or often makes it outright possible to do at all). Then if you apply integration, which is the opposite of differentiation like how division is the opposite of multiplication, to expand it to a more complex whole.
The most simple examples of this tends to be calculating areas and volumes of shapes. Things like spheres, circles, cylinders, cones, prisms, pyramids, or a funny wave cylinder formed by rotating a sine-wave around lathe. if it were on a lathe. These can all be derived using integration of infinitesmal elements. But can be used for much, much more like calculations of fields in fluid dynamics, electromagnetics, flows, heat, and all sorts of other weird stuff.
For example, to calculate the volume of an arbitrary geometric shape you start by describing what the volume of a piece of this shape is. As you move from piece to piece, the volume of the piece changes a bit. That could be described as a rate of change but it is most useful to think of it as describing a tiny piece of the whole. You can then integrate all the tiny pieces to get the volume of the whole. This is integration and you asked about differentiation but it is difficult to integrate if you don't know how to differentiate.
One example is area of a pizza using slices. The area of a circle is difficult if you don't already know the equation. The area of a large pizza slice is also difficult since it's almost like a triangle where one side has a bit of a curve, But if you use an infinitely small slice that curved side is essentially straight and you can just treat it as a triangle. The area of a triangle is easy. Then you can integrate up a bunch of infinitesmal triangular slices rotating them around the center of the pizza to get the area of the whole. The rate of change here is the additional volume the pizza gets for every additional infinitesmal slice that you add.
Differentiation tends to be most direclty useful in differential equations. You may have only encountered equations so far where the output depends on the input but in general, things depend on the input and how fast the input is changing and how fast the output is changing. The most classic example is the heat equation. The temperature of an object stabilizes when the rate of heat flowing into an object equals the rate of heat flowing of an object, and the rate that heat flows into or out of an object depends on the difference in temperature between the object and its surroundings.
A: 
why does differentiation has so much significance in physics, if we are only dealing inf. small changes.

Dealing with infinitesimally small values can be overwhelming for beginners. I suggest you take a good book on calculus and invest a lot of effort/time in studying it, as calculus is very important to understand many concepts in physics.
I will just mention here that an infinitesimally small variable on its own can be neglected, but ratio between two infinitesimally small variables cannot be neglected and has to be handled with care.

what does it mean to find the rate of change of a quantity at a particular value (K here)?

The volume rate of change is not constant, but is a function of the sphere radius. The problem you stated asks you to find a numeric value of the volume rate of change at one specific radius.

The volume of a sphere as a function of time-changing radius is
$$V(t) = \frac{4 \pi}{3} r^3(t)$$
The time-derivative of the sphere volume is
$$\frac{d}{dt} V(t) = \frac{4 \pi}{3} 3 r^2(t) \frac{d}{dt} r(t) = 4 \pi r^2(t) \frac{d}{dt} r(t)$$
The following notation is often used for time derivatives
$$\frac{d}{dt} x(t) \equiv \dot{x} \qquad \text{and} \qquad \frac{d^2}{dt^2} x(t) \equiv \ddot{x} \qquad \text{etc.}$$
The above equation can then be written as:
$$\dot{V} = 4 \pi r^2 \dot{r}$$
where $\dot{r}$ is the rate of change of the sphere radius ($1 \text{ m/s}$). The question asks to find $\dot{V}$ when $r = \frac{1}{\sqrt{\pi}}$ which is
$$\dot{V} = 4 \pi \cdot \frac{1}{\pi} \cdot 1 = 4 \text{ m}^3/\text{s}$$
In your question you actually specify to find $\dot{V}$ when the volume equals $\frac{1}{\sqrt{\pi}}$. If that is what you want, you first find radius for the given volume and then repeat the steps above for new radius. In this case the correct answer is not $4 \text{ m}^3/\text{s}$.
A: 
So why does differentiation has so much significance in physics,

Because it's one of the most fundamental mechanisms in nature. Many physical phenomena are caused by the CHANGE of a another physical thing (and not the thing itself). Differentiation is the mathematical process to quantify the amount or rate of change.
Let's look at a loudspeaker cone moving. The CHANGE in position moves the air. The CHANGE in air movement creates a pressure and so we get sound.
Light is basically caused by a CHANGE of electric field which causes a CHANGE in the magnetic field etc.
When you move from A to B, the velocity is the amount of CHANGE in your position. The more your position changes per unit of time, the higher your speed is.

As far as I understand dx is very small change(infinitely small?) in x

This is really just mathematical tool to quantify the rate of change. Rate of change is easy if have for example linear motion (10m after 1 s, 20m after 2s, etc.) but the formalism with the dx works for any type of function.
A: Infinitesimals have been very important for understanding motion for a very long time.  You may have heard of the paradox of Achilles and the Tortoise.  It's Zeno's most famous paradox.  The idea is that Achilles and the Tortoise are in a race.  The Tortoise is given a head start.  Zeno argued that to catch the Tortoise, Achilles must first run to where the Tortoise is at the start of Achilles' run.  But by the time he gets there, the Tortoise will have moved further.  So Achilles must run to that point.  But when he gets there, the Tortoise has moved even further.  And so on and so on.  Zeno this argues that, by the understanding of motion at the time, Achilles will never catch the Tortoise.
Now we know this to be absurd.  We all know  that Achilles will eventually overtake the Tortoise.  But how do you explain that logically.  By Zeno's argument, there's an infinite number of steps for Achilles before overtaking the Tortoise.  They get infinitely small, but infinite in number.  What are we to do?
The calculus invented by Newton and Leibniz is "the calculus of infinitesimals.". A calculus is just a method of calculating something algorithmically.  This calculus of infinitesimals was the first consistent way of dealing with an infinite number of infinitesimals.  This was the first resolution of Zeno's paradox by rigorously showing what happens as Achilles moves towards the Tortoise.
This turns out to be such a big deal that its name got shortened from "the calculus of infinitesimals" to just "Calculus.". What it lets us do is make huge statements about the continuum that is the real number line.  It frees us greatly from having to limit ourselves to rational numbers.
When you ask about the infinitesimals rate of change, like you do in your question, you are getting at the heart of what is needed to sum together an infinite number of them.  Derivatives are a powerful concept on their own, but where they really shine is when combined with the next thing you will learn, integration, which lets you take an infinite number of these infinitesimals (from taking derivatives) and sum them together sanely to get a finite result.
As a concrete example, consider a problem involving the distance a car travels at a constant speed over a given time.  You can solve this one without calculus.  It's a special case where a few triangles and some multiplication is all that is needed.  But what if the car isn't at a constant speed?  What if it accelerates from a standstill?  What we find is that it's instantaneous rate of change in position (otherwise called "instantaneous speed") at any given time is the fundamental building block you need to solve the more general "how far did it go" questions.
As a specific example, consider the famous equation for motion of an object with a constant acceleration: $x=\frac 1 2 a t^2 + vt + x_0$.  You could drive this through a lot of experimentation, and find the equation empirically.  With calculus, we only need $\frac{d^2x}{dt^2}=a$, which you will learn is the notation we use for a constant acceleration (the instantaneous rate of change in the instantaneous rate of change of position), and the fundamental theorem of calculus, and that original equation just pops out.
This sort of thing lets us describe motion in more complicated situation.  Consider the motion of satellites.  We observed that satellites (such as the planets) orbit in ellipses in general.  However, it takes calculus to figure out how to change one orbit into another.  This ends up being essential for things like planning a rendezvous between your Dragon capsule full of astronauts and the ISS that they are trying to reach
A: Note this is very slack language, and not precise. It's to give a flavour of the answer, not to be exact technically.
The big deal of it, is really in three parts.

*

*A lot of calculations are really complicated with basic mathematics, or just plain impossible. We need new ways to do many "real world" problems that bypass these difficulties. 

*A huge number of problems, are difficult because they involve something that varies gradually. A curve of some kind, or something that's not a straight line (for example) if you drew it. But if you looked at the tricky curve with a bigger and bigger magnifying glass, focusing on smaller and smaller bits of it, it would start to look more and more almost like a straight line.We can do maths on straight line things much easier. So if we could imagine extending that with an infinitely powerful magnifying glass, we would get more and more, flatter and flatter line segments.Differentiation and integration are words we use to describe this process. We can  break down a shape or curved thing, mathematically, into a vast number (infinite number) of tiniest flat things (infinitesimally small items). We can also add up this infinity of flat parts muxh easier, to "reassemble" the curve ...... but this time in a way that we can use maths of straight lines to do the hard work, bypassing the original roadblock. 

*It also turns out when we do this, we have found ways to analyse how things change, and how things build up, which is a hugely important trick with an incredible number of practical uses in mathematics, physics, and the real world.(Will this beam support that weight? How much water to fill this odd shape container? How much fuel is needed to get a satellite to orbit, given it has to use more fuel to carry that fuel? How fast or far will we travel? What is the point on a curve that is lowest or highest? Literally, more uses than you could count in a lifetime. Those are some simple ones.)

Update - your last point
I didn't really answer this, and want to add a simple example:

why does differentiation have so much significance in physics, if we are only dealing with infinitesimal small changes? Also, what does it mean to find the rate of change of a quantity at a particular value (K here)? What’s the use?

Suppose an object moves around a circular path 10m wide (diameter), at 1 m/sec. So in an ideal world of mathematics and physics problems, it has a circumference of 31.4m, and takes 31.4 seconds to walk around, or about 7.9 seconds to make a quarter turn.
Now, if we scale that down, that means in 7.9 seconds we turn 90 degrees. So in 1 second it turns 11.6 degrees. In  1millisecond it turns 0.0116 degrees, and every microsecond, turns just 0.0000116 degrees which is a tiny amount.
The point I'm making here is that the numbers are becoming infinitesimal, but you cant ignore the rate of change. 360 degrees in 31.4 seconds is the same amount of turning as 90 degrees I  7.85 seconds, or 11.6 degrees in 1 second, or 0.0000116 degrees in one microsecond.
That's the "rate of change". The rate of change here, is how quickly its direction changes per unit time - and that's the same rate whether you say 11.6 degrees in 1 second, 1.16 degrees in 0.1 seconds, or 0.0116 degrees in 1 millisecond.
So the rate of change of the direction it's moving, is the same for all of these.
In a way its the same as saying 2/4 = 3/6 = 1000/2000. The ratio is what matters, not the tiny sizes. That's an important insight.
Like I said above, what the tiny sizes let us do, is to approximate tricky mathematics about curves and shapes, by tiny straight lines that are much easier to work with.
