Problem in understanding the concept of differentiation. I have just completed the chapter differentiation. But I still have confusion in understanding the concept of it.I observe a fact that if we can draw a tangent to a curve at any point of it or in other words if it is possible to make the best linear approximation to a curve at any point then it is said that the curve or equivalently the function is differentiable at that point.But I fail to relate this concept to various other facts of differentiability like velocity or acceleration.Why do we consider the best linear approximation of the displacement curve, traversed by a particle, at any time t as the velocity of the particle at time t. I have similar confusion regarding acceleration. Please help me in understanding these important facts reminding that I am a new to this subject.Thank you in advance.
 A: We relate derivation to motion with respect to time, because these ARE derivatives.
To understand it first let's draw a graph where on the y axis is distance, and on x axis is time. Let's say something is moving in a constant speed. The line we would draw on the distance-time graph would be a straight line in the form of $ax+b$. If we take a derivative to find velocity at a certain point in the distance-time graph, we will see, that we will get just $a$, which is constant just as we would expect from a velocity of an object with a linear distance-time graph.
On the other hand, if we put a distance-time graph to be a function like $x^2$, to calculate svelocity at a certain point, it will be harder to do. To calculate it we would have to pick another point, calculate the change in y, and divide by change in x, which is exactly the definition of derivative. As you may know from physics, the formula for velocity is ending point mins starting point divided by time:
$$v=\frac{d_{final}-d_{initial}}{t}$$
But hen you have for example $x^2$ as your graph, velocity would not be exact, so we add an limit if inititial distance, as well as time, to approach 0. This is definition of derivative RIGHT THERE. We approach 0 for both init distance and time because time is just change in x. Initial distance is also change in x from the final distance. So thinking deeper, it is almost same thing, even when it is not.
So to calculate velocity at a certain point of a distance-time graph, we would have to use derivative, otherwise we would get results that are not exact. If you draw an example graph on paper and graph a function on it, and try to figure out speed at certain point, you will see why we use derivatives for this.
Same applies for higher orders like acceleration. Let's bring back the linear distance-time graph. It is in form of $ax+b$. If we take a derivative of this, we will get constant $a$ which would be a constant velocity. From physics, we know that when speed/velocity is constant, there is no acceleration. When we take the derivative of our constant velocity of $a$, you will see we get 0, which informally proves that derivatives are present in physics of motion.
EDIT: and if I'm not wrong, derivatives were most probably discovered because of physics of motion rather than properties of physics of motion discovered because of derivatives. Also Newton was one of the inventor of calculus, and he made up all the most important physics laws, so you can see the connection between those two discoveries.
A: The derivative of $f$ at time $t$ is defined as $f'(t) = \lim_{h \to 0} \frac{f(t + h) - f(t)}{h}$.  Here are two ways to think about this definition.  Viewpoint 1: $\frac{f(t+h) - f(t)}{h}$ is the average rate of change of $f$ over the interval $[t,t+h]$.  The average rate of change approaches the instantaneous rate of change as $h \to 0$.  Viewpoint 2: $\frac{f(t+h) - f(t)}{h}$ is the slope of the secant line connecting $(t,f(t))$ and $(t+h,f(t+h))$.  The slope of the secant line approaches the slope of the tangent line as $h \to 0$.
A: You probably know the concept of slope. Its formula is $\frac{y_a-y_b}{a-b}$, where $a$ and $b$ are some values on the $x$-axis, and $y$ is the function. A derivative is the idea that the slope off of any continuous function can be found. This is defined as $$\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$$ As $h$ gets smaller and smaller, it becomes more and more accurate. It is useful because it can help find the velocities and accelerations you spoke of. The slope of the displacement function is the velocity, and the slope of the velocity is the acceleration.
