There are two ways in which a derivative can be defined. Geometrically it can be defined as the slope in a curve and physically it can be defined as the rate of change.
It can be represented as the ratio of change in the functional value to the change in a variable. It can measure graph steepness concerning a specific point in it if X and Y are the coordinates of a graph. Slope = change in Y/change in X.
The physical concept of a derivative:
This concept was used by Newton in classical mechanics and used it to define speed and velocity. Assuming you are traveling from point A to B, to calculate the average velocity in that trip:
Average velocity = distance from A to B/ time taken to cover the distance.
If A and B are close to each other, then the above formula can be used to calculate the instantaneous velocity. Moreover, if the distance from A to B is less than the time taken to cover the distance is also small.
The geometrical concept of a derivative:
It is derived as the rate of change or slope at a particular point of a function. On the curve, you can find the slope at a specific point. It is represented by ‘dy/dx’ which is the symbol of the derivative of y with the derivative to x.
Some real-life applications of derivatives:
- It can help you run and grow your business: Consider you are in the business of ice-cream selling. The derivatives can help you calculate the quantity to sell to make a profit. Based on the data of how the business is doing in the past months you can use a graph to determine the trend. If you are selling every ice-cream for $10.
Plot the cost on the y-axis and the produced units in the x-axis. That shows the growth in your business. As the quantity increases, the curve becomes steep, and it can be because as you produce more ice-cream, there will be other costs that will make the curve steep.
The derivative which is ‘dy/dx’ gives a slope which provides you with the cost of producing additional units. It helps you decide on how much ice-creams you should produce.
- While playing games: When you are playing a shooting game making the right predictions on the enemies move is essential as it helps you shoot down the player. The prediction is made based on the direction and the speed which is the derivative of the player’s movement. You decide where the player ends up with the help of mental integration which you do it without even knowing about it and is thus one of the best real-life application of derivatives.
- When it comes to the census, the population is a derivative to time and is proportional. Some systems might be more complex, the time derivative of the population is proportional to the difference in the population and the sustainable population factored by the population.
Derivatives are very useful and much longer, as they represent slope and they can be useful to find different functions and they have different uses in the field of physics and optimization. Climate changes are also determined, a scientist can calculate the changes and melt of icecaps, so derivatives are useful in different ways.