I want to know what is the use of derivatives in our daily life.
I have searched it on google but i haven't find any accurate answer. I think it is mostly used in Maths but I want to know its use in other departments i.e physics, chemistry, biology and economics.
On
All of the above. It is actually easier to explain physics, chemistry, economonics, etc with calculus than without it.
For example:
Velocity is derivative of position with time.
Derivative of momentum (by time) is force.
Derivative of Gibbs free energy with number of atoms is chemical potential. Etc.
I do honestly use calculus daily and I am not a mathematician. Please feel free to add to this list.
On
Physics: A typical use of the derivative is this analysis of air pressure in the atmosphere as a function of height. You say "Let's look at a column of air. At height $x$ there's some pressure $p(x)$, and the air at that height is compressed in proportion to $p$, i.e., the air density is a constant multiple of $p(x)$. The pressure $p(x)$ is related to the total weight of all the air above height $x$ -- it's some multiple of that number. If we assume that our column of air is 1 cm square, then the pressure, in grams per square centimeter, is exactly the weight of the air above the height $x$.
So where do derivatives come in? Let's look at the pressure at some height $x$ and at height $x + h$ (for some small value of $h$) and ask how they differ. The only difference is that at height $x$, we're supporting all the air from $x$ to the top of the column, which we can divide in two parts: the air between $x$ and $x+h$, and the air from $x+h$ to the top of the column. If we write $w(x)$ for the weight of the air above height $x$, then I've just said that $$ w(x) = w(x + h) + \text{weight of all air in the column between $x$ and $x+h$}. $$
The weight of the air is the density times the volume; the density is proportional to the pressure, which is in turn proportional to the weight. I'm going to say that $h$ is so small that the density between height $x$ and height $x+h$ is pretty much a constant, so I can say it's proportional to $w(x)$ times the cross-sectional area of the block (1 square cm) times the height of the block ($h$). So my formula becomes $$ w(x) = w(x + h) + C \cdot w(x) \cdot h. $$ With a little algebra, this turns into $$ - C \cdot w(x) \cdot h = w(x + h) - w(x), \text{or} \\ - C \cdot w(x) = \frac{w(x + h) - w(x)}{h}. $$ That thing on the right, as $h$ gets smaller and smaller, turns into the derivative, $w'(x)$. So we get an equation for $w$: we're looking for a function with
$$ - C \cdot w(x) = w'(x). $$
See? There's a derivative right there. And once you know how to solve equations like this -- which you'll learn when you get to integration -- you're got a way to compute how air pressure changes with height.
Chemistry: The rate at which a reaction takes place is typically proportional to the concentrations of the constituents and products. If your reaction consumes $A$ and $B$ and produces $C$, then high concentrations of $A$ and $B$ lead to fast reactions, but high concentrations of $C$ may drive the reaction backwards. Sometimes you can have reactions in which raising a concentration of some chemical inhibits the reaction. Now imagine you have two reactions going on at once, and each produces a product that inhibits the other reaction. Let's also assume that this process is taking place in something like a jelly, so that reactants and products "diffuse" throughout the jelly. Then if you start with a random distribution of reactants, they react and produce inhibitors, which then diffuse, and...well, how can you describe this or predict what'll happen? Well, the rate of reaction is expressed with a derivative, and the process of diffusion is described with derivatives, and by combining these you get an equation -- like the one for air pressure, but more complex -- whose solution describes the eventual state of your chemicals. By the way, this model of "reaction-diffusion" is believed to be a possible explanation for patterns like leopard spots: the reactants diffuse through the tissue, and the concentration of SOME reactant determines whether the hair grows yellow or black, etc.
Economics: The classic example is a model economy that's very simple: you've got two factories, one making guns, one making butter. You can put all your workers int he gun factory, and get no butter, or all in the butter factory and get no guns, or some in each. The set of possible outputs is called the "production possibility curve." Now there's also a need for guns and for butter. If there are too few guns, the demand for them will be high; if too little butter, the demand for butter will be high. Considering all possibilities, you get a "demand curve". And the economy is "tuned just right" when the production is set up to exactly match the demand, i.e., when producing a little more butter and a little less armament would result in net dissatisfaction (the butter-using folks like it, but the gun-users hate it even more) and when producing a little less butter and more guns would lead to a similar increase in net dissatisfaction. How do you find that magic point? Turns out that you do it by setting a couple of derivatives to be proportional to one another and solving an equation. The results you get from this example can then be used to help understand more complex economies as well (or so they tell us).
"... the drop in the economic growth rate decreased..."
may not be worth a headline, but it's an example of third derivatives within economics.