Uses of derivatives in Calculus All introductory books on calculus talk a lot about how to get derivatives of functions, this is fine with me. What however that's still confusing to me is the real use of derivatives.
for example the y(x) = x^2 has 2x has its derivative.
How is 2x useful in relation to the original function.
 A: Note first that pure mathematics is silent on the real-world application of their concepts. And pure math is not obliged to propose any real-world application to anybody. I recall that there is an anecdote about Richard Feynman. Young Feynman was not sure about whether to study math or physics. After being asked about how useful mathematics is, the mathematics professor Feynman consulted told him something like "if you need to know how useful mathematics is in order to study math, then you are probably not suitable for pursuing an academic career in math". The point here is not about who is suitable for studying what; the point is about the math professor really told a truth about a very characteristic of math!
The problem of finding derivatives may date back to the problem of finding the slope of the tangent line to a given curve at a given point. Now if the curve is the graph of the function that gives us a position of an object at any given time, then, given any time, the slope of the tangent line to the graph of the function at the given time is interpreted as the velocity of the object at the given time. Here the mathematical objects such as functions or derivatives are employed as a model. In fact, you have already accepted the concept of using mathematical objects as a model for studying the real-world phenomena, i.e. the use of analytic geometry in physics. For instance, have not you seen in your mind that the graph of the function in the above example lies on a two-dimensional plane with the Cartesian coordinate axes on it? If you accepted the use of analytic geometry without a problem, then in the same token you can accept the use of derivatives. As it seems to me, it is just that you were somehow scared by the concept of derivatives so that you stopped and saw if you can find some applications of that concept to make it more real to you. If this is the case, please do not continue; make the concept more real to you by playing with it and trying to make the concept of limit, which is more fundamental than that of derivatives, more real to you first.
A: Derivatives are useful for describing the ways that functions evolve. Some examples of uses are listed below;


*

*Can provide information to generate a tangent to a graph in the x-y plane 

*Can be used to compare the effects of different variables on a function

*Find stationary points of a function e.g maxima, minima, saddle points. These allow for the values that optimise functions to be found, e.g maximising the size of a rectangle that can be drawn within an ellipse

*Show the vector of greatest increase in a scalar field, if for example you wanted to walk up the steepest part of a hill


There are many other uses of derivatives, but fundamentally, it can be very useful to understand how a function is changing under specific conditions.
A: Note that $f'(t)$ is the instantaneous rate of change of $f$ at $t$. For example, if $f(t)$ is the distance a car has traveled along a road at time $t$, then $f'(t)$ is the car's velocity at time $t$. (And you can find out the value of $f'(t)$ by looking at the speedometer.)
The average rate of change of $f$ over the interval $[t,t+\Delta t]$ is
$$
\text{average rate of change} = \frac{f(t+\Delta t)-f(t)}{\Delta t}.
$$
Take the limit of this average rate of change as $\Delta t$ approaches $0$ to obtain the instantaneous rate of change of $f$ at time $t$:
$$
\text{instantaneous rate of change} = \lim_{\Delta t \to 0} \frac{f(t+\Delta t)-
f(t)}{\Delta t}.
$$
