Have you ever heard of the term derivative? Well! It’s a fundamental tool of calculus. What makes it unique, is the fact that this tool can compute the change of a function at any point. In calculus, this concept is equally important as integral, which is the reverse of derivative also called anti-derivative.
The rate of change concept, makes it a valuable asset in many real life applications. For instance, the diversity of temperature can be checked using this notion. In this article, we will discuss in detail, its definition along with the real life utility.
Now let’s get started, at first we will try to understand the concepts of derivative and differentiation.
Definition:
The derivative of a variable is defined as a measure to compute the rate of change of a function’s output value as it varies from the initial value or input. Here, an important thing is the time factor, the variation in input and output value as time changes.
Let’s consider an example of a moving object, the location of that object starting from the initial point, with respect to time is considered as object’s velocity. This tells us the relative swiftness of the object as it deviates from its position, as time advances.
Here, the image above, illustrates a tangent line. The slope of the tangent line at the marked point represents the derivative of a function. The variation can be projected by the ratio of change of function Y (dependent variable) to that of the variable x (independent variable).
Notation:
A German mathematician, Gottfried Wilhelm Leibniz’s introduced a notation, in which symbols were given; dx, dy, and dy/dx. It is commonly used in case an equation y=f(x) is viewed as an association of dependent and independent variables.
It uses these symbols to define the infinitesimal (very small) increments. On the other hand, symbols such as Δx and Δy are used to represent the finite increments of x and y.
Differentiation:
It is a process that helps in calculating the derivative, just like integration computes an integral. This operation is reverse of integration.
Let’s assume y a linear function of x. In this example, y = f(x) = mx + b, let m and b the real numbers, slope m is expressed as
Slope = m = change in y / change in x = Δy/ Δx
Here, Δy = f(x + Δx) – f(x), the above equation is because;
= y+ Δy = f(x + Δx)
=m(x + Δx) + b = mx + mΔx + b = y + mΔx
This gives us the slope of line, Δy = mΔx
It applies to a straight line, if the graph is not linear, then the change varies over a considerable range. The differentiation is an efficient method to compute this change over a specific value of x.
Practical Applications:
This tool isn’t just limited to mathematical problems, it has a broad range of practical utility. Nothing is useless in this world, when we say something can’t be used, we actually don’t know how to use it. The one who knows its utility, won’t stop thinking about it.
The uniqueness of this concept is its predictive ability to evaluate the change in quantities. Whether its speed, momentum, temperature and even the business speculations, all the variations can be worked out using derivative.
Use in Physics:
As we mentioned above, the example of a moving body’s relative position can help us calculate the velocity.
In the same way, derivatives of acceleration and momentum can be found.
Use in Chemistry:
In chemistry, the concentration of an element involved in a reaction, the change in concentration can be predicted.
Similarly, to measure the rate of chemical reactions and to check the contribution and loss of a compound during the reaction.
Use in Economics:
Nowadays, the decision making in economics has become more mathematical. Statistical and mathematical principles are applied in making decisions regarding possible gain or loss in investment.
Confronted with massive statistical data, dependent on lots of variables, there was a need of some tool that could assist the analysts.
Here, calculus proved to be beneficial. It implemented the derivative concepts to predict the results of different investment possibilities.
Ultimately, this enabled the analysts to select the one possibility that might prove to be productive in terms of profitability.
In the end, I hope this article will help you understand and apply the calculus concepts in practical fields. If you are interested in methods to calculate this fundamental of calculus, try this derivative calculator. You can also make a relevant calculation on integral function on this integral calculator.