## Physics

The derivative plays a gigantic role in physics. The concept of instantaneous activity is of utmost importance-- using algebra, we can calculate the average speed of an object, yet, with the derivative, we can calculate the instantaneous speed (the speed at any given instant) of an object.

It works like this. Think about a car moving along the x axis. If the location function, or the distance function (the car is f(t) away from the origin after time t) of the car is f(t), then its instantaneous velocity at t is just f'(t). Why does it works like this?

So

Probably you have seen distance-time graphs. And the slope=velocity. It is the same here. Think of the derivative=slope of tangent line=slope of a point.

Another concept that comes in is acceleration. Everybody knows that F=ma. But what is a? Suppose we go back to our definition of the "position function", f(t). Take a guess using the definition of acceleration.

If your guess is not too far off, it should read a=f''(t). That is, the acceleration is the second derivative of the position function, the derivative of the velocity function.

It works like this. Think about a car moving along the x axis. If the location function, or the distance function (the car is f(t) away from the origin after time t) of the car is f(t), then its instantaneous velocity at t is just f'(t). Why does it works like this?

So

*.*__v(t)=f'(t)__Probably you have seen distance-time graphs. And the slope=velocity. It is the same here. Think of the derivative=slope of tangent line=slope of a point.

Another concept that comes in is acceleration. Everybody knows that F=ma. But what is a? Suppose we go back to our definition of the "position function", f(t). Take a guess using the definition of acceleration.

__a(t)=f''(t)____.__If your guess is not too far off, it should read a=f''(t). That is, the acceleration is the second derivative of the position function, the derivative of the velocity function.

## Economics

The derivative also has significant appliances in the field of economics. Setting complicated models aside, we can look at the simple concept of "marginal benefit". Or "marginal utility".

Well, for those of you who are not familiar, marginal benefit is just a fancy way for economists to say the happiness you get from doing *something* a little bit more. So, perhaps it's a car business. Then, the marginal benefit is the profit you get from selling another car. Yeah?

So now, we have the profit function f(t). Don't ask me how do we get it. That's usually the difficult part. But what comes up in tests is ONCE WE HAVE f(t), what is the marginal benefit at say t=1. (Only selling one car makes one a terrible salesman.) Now that would just be f'(1).

By the way, there are something in economics called the Law of Diminishing Returns, stating that once something has been done too much, the benefits start to decline. Think about eating stuffed chocolate cakes. It only takes so many to make one puke.

Well, for those of you who are not familiar, marginal benefit is just a fancy way for economists to say the happiness you get from doing *something* a little bit more. So, perhaps it's a car business. Then, the marginal benefit is the profit you get from selling another car. Yeah?

So now, we have the profit function f(t). Don't ask me how do we get it. That's usually the difficult part. But what comes up in tests is ONCE WE HAVE f(t), what is the marginal benefit at say t=1. (Only selling one car makes one a terrible salesman.) Now that would just be f'(1).

By the way, there are something in economics called the Law of Diminishing Returns, stating that once something has been done too much, the benefits start to decline. Think about eating stuffed chocolate cakes. It only takes so many to make one puke.

## Something Else to Say

I would not bother with the other myriads of applications because there are so many. It depends on the field in which you wish to study. Chemistry also makes extensive use of calculus. Calculus is used to explain reaction rates, ideal gas expansion, among many other things. Perhaps you have heard the logistic curve. Biologists use it to account for population growth in an environment. The curve came out of a differential equation. Calculus is the basic math requirement if you wish to pursue a career in science.

Even though nowadays, there are computer softwares that can help you do all the math calculations, an understanding of the principles and ideas at work behind the calculations is critical if you wish to apply mathematical models to the real world.

Even though nowadays, there are computer softwares that can help you do all the math calculations, an understanding of the principles and ideas at work behind the calculations is critical if you wish to apply mathematical models to the real world.