what is the use of derivatives

My example is from the real life situation of war. From experiments in physics we know that the acceleration due to gravity of a particle near the Earth's surface is about $-9.8 \frac{m}{s^2}$.

If you're an artilleryman in an army, you want your artillery shells to hit the enemy, or else theirs may hit you and kill you (game over). So you need to know how to angle your artillerygun and which direction to point it in so that when the shell lands, it blows up your enemy (rather than missing).

All you know is that you can control the direction your cannon points and the angle you fire it in in the air -- after it's fired gravity takes over and that $-9.8 \frac{m}{s^2}$ takes over the situation.

Calculus shows us that if $x(t)$ is the function representing the position of the artillery shell at time $t$, then its first derivative is the shell's velocity and its second derivative is the shell's acceleration. We know acceleration from physics (it's that $-9.8 \frac{m}{s^2}$ we had earlier)! We write this as $x''(t) = -9.8 \frac{m}{s^2}$.

From this equation (involving derivatives) you can calculate (using "integration") the position function $x(t)$ of the particle given the direction you fire it in and the angle you fire it at.

Why do you care about that? Because knowing $x(t)$ will tell you where your shell lands and thus whether your shot will kill the enemy or not. So you can do a quick calculation to determine which direction and which angle to fire in to ensure that your shell hits your target. The side that does this computation first and gets the shell in the air first will kill the other side, helping win the war.

Yes, the derivative can be used to determine the "rate of change" but more generally can be viewed as a tool to approximate nonlinear functions locally with linear functions. This is true in the case of a real-valued function of a real variable and is the case in higher dimensions such as a surface defined by a multivariable function. In the former case, the linear function is realized as the line tangent at a given point and in the latter as the tangent plane. The derivative is, in essence, the best linear model available for a function in a neighborhood of a point.

The derivative, by providing a mechanism of "local linearization", can turn a hard/intractable problem into a problem of linear algebra which is usually easier to deal with. In real life, the utility comes into play when complex behavior of physical systems is modeled by nonlinear functions which can in then in turn be locally approximated by the derivative.

From another point of view, the derivative represents how one quantity changes as another quantity varies. In many cases, we can construct models by relating quantities to their rates of change, then using tools from differential equations to make predictions. The predictions are usually very accurate, which is why we teach calculus to our high school and college students.

Examples:

A simple object's position is governed by an equation relating the second derivative of its position to the forces on the object: $F_{net} = mx''(t)$.

Short-run employment in the economy depends linearly on the time derivative of the difference between the expected price level and the actual price level.

The rate at which heat spreads through an object from a constant-temperature source is related to the derivative of temperature with respect to distance.

In simple biological systems, the population is proportional to its time derivative. In slightly more complicated biological systems, the time derivative of the population is proportional to the population multiplied by the difference between the total sustainable population and the population.

This list could be much, much longer, but I think it illustrates the point. Derivatives are useful.

Derivatives are very useful. Because they represent slope, they can be used to find maxima and minima of functions (i.e. when the derivative, or slope, is zero). This is useful in optimization.

Derivatives can be used to estimate functions, to create infinite series. They can be used to describe how much a function is changing - if a function is increasing or decreasing, and by how much. They also have loads of uses in physics. Derivatives are used in L'Hôpital's rule to evaluate limits. Derivatives can even help you graph a function!