Calculus For Everyone

The Fundamentals Of Derivative

Derivative — definition and formula description

The world is always changing. When the car is running, the position of the car changes. As water evaporates, the volume of water changes. Even when you play games, the position of your game character also changes. There are so many changes happening in this world, and they happen all the time.

As I said in the introduction, calculus is the study of change. A branch of calculus — that is derivation—studies how fast a change occurs. Take the example of a moving car. In that case, the derivative learns how fast the car’s position is changing. In other words, the derivative studies the speed at which the car is moving.

if the car was moving at a constant speed, it would be very easy to calculate its speed. Speed ​​is equal to the distance traveled (change in position) divided by the time traveled. This is very easy. If depicted, the graph for a car with constant speed is as follows:

To calculate the speed of the car, we only need to calculate the distance from the start point and endpoint, then the distance is divided by the travel time.

Based on the graph above, we can calculate the speed of the car by dividing ∆s by ∆t. So, v (velocity) is equal to ∆s/∆t.

What if the speed of the car is not constant, but changing? As we know — in the real world — the speed of a car is not always constant. Sometimes it accelerates and slows down. sometimes the car can even move backward. That way, the relationship between changes in the position (distance) of the car to travel time can form the following graph:

If the graph is like the picture above, how do we know the speed of the car? Can we use that method again? Let’s try.

By calculating the difference in the value of s between point A and point B, we get the value of ∆s. By calculating the travel time from A to B, we get the value of ∆t. in the same way as the previous example, we can calculate the speed of the car by dividing ∆s by ∆t. In other words, v=∆s/∆t.

Oops, wait. Is this true? The value we calculated earlier is only the average speed of the car from point A to point B, not the real speed. If the car’s speed is constant, its average speed becomes its real speed. However, this does not apply to cars with non-constant speed. In a car whose speed is not constant, it could be that at t=1 second the speed is 10 m/s, but at t=3 seconds, the speed becomes 20 m/s. That is, the speed of the car can change every second. The speed at a certain moment is called the instantaneous speed.

Then, if the above method cannot be used to calculate the speed of the car, what method should we use?

Take a look at the graph again. Then, ask the question, what is the speed of the car at t = 4 seconds? We can calculate it using a method similar to the previous method, but only with a very short distance (∆s) and short time span (∆t). Take a look at the following graph:

Try to imagine points A and B moving closer to each other continuously like the three pictures above. Points A and B keep getting closer until ∆s is very, very small. We can view this as follows:

When points A and B approach each other until they almost touch, the graph connecting point A and point B becomes very similar to a straight line. Thus, we can calculate the speed of the car by calculating the gradient of the line for ∆t close to 0. Thus, the speed of the car can be denoted as the limit of ∆s/∆t with ∆t approaching 0.

Besides being able to calculate the instantaneous speed of a car, the above formula can calculate the instantaneous speed of everything that changes. For example, changes in the price of goods due to demand, changes in the length of objects due to expansion, and others.

General Formula

In the discussion above, we only talked about the speed of the car. Now, we will discuss the concept of speed in general. As in the example above, velocity can be calculated by dividing ∆s (the y-axis on the graph) by ∆t (the x-axis on the graph). In mathematics, ∆s/∆t or ∆y/∆x is called the gradient or slope.

The slope of the function f(x) describes the rate at which the value of f(x) changes on the graph. So, it can be said that the change in the value of the function f(x) with respect to x is equal to the slope of the function f(x).

The general formula for calculating the slope of a function on a graph is as follows:

Now, if we take a very small value of ∆x, then this gradient is called the first derivative of the function f(x). The first derivative of the function f(x) is denoted as f ' (x).

Conclusion

Derivatives study how quickly a change occurs. To track these changes, the derivative uses the concept of limits that we studied earlier. Basically, the concept of a derivative is the same as the concept of a slope. However, the derivative is much more useful because it can calculate the slope of the function under any circumstances.

Thank you for reading this article! If you still don’t understand my explanation, don’t hesitate to ask in the response section. I understand that my writing is quite complex, so it’s a bit difficult to explain it only through writing and pictures. Therefore, please ask if you do not understand my explanation.

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