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Newspaper headlines are often filled with global issues such as climate change and the greenhouse effect, which naturally raise questions like: How can we model changes in sea levels caused by these factors? At what temperature will glaciers begin to melt at a certain rate? How does the number of Covid-19 cases evolve a month after an outbreak begins?
The answers to such questions lie in mathematical analysis, particularly in the study of tangents and rates of change — problems solved by derivatives.
When we look at the broad field of mathematical analysis, we might agree it is not always easy to write or speak about its principles in a way that resonates with everyone. That is why, in this article, we will approach the concept of derivatives in a slightly (non)formal and practical way.
To complete one’s understanding of calculus, it is necessary to grasp three key pillars: limits, derivatives, and integrals (the latter being essentially antiderivatives). Just as everything in life follows an order, so does mathematics: in order to understand derivatives, one must first understand limits. However, we will not focus here on rigorous proofs or theorems. Instead, we will look at why derivatives were discovered, how they are applied in everyday life and science, and give just a brief overview of limits where needed.
Two mathematicians and physicists — the Englishman Isaac Newton and the German Gottfried W. Leibniz — independently developed differential and integral calculus around the same time. Both were motivated by physical problems: Newton by the study of instantaneous velocity (1666), and Leibniz by the problem of finding tangents to curves (1674).
Although Newton studied the subject earlier, Leibniz was the first to publish his findings. This sparked not only personal rivalry but also a national one, as England and Germany defended their champions.
What is important here is that derivatives were not born from abstract speculation but as a mathematical tool for solving physical problems. Only later did they take their place in higher mathematics, and today they underpin countless independent branches of science.
A limit describes the value a function approaches as its input nears a particular point. Once limits are established, derivatives can be defined. In general, derivative describes how fast something is changing at a specific point. Everyday examples:
If the function is distance traveled over time, the derivative is speed (how fast you’re moving at that exact moment).
If the function is temperature during the day, the derivative tells you how quickly it’s warming up or cooling down at that moment
If a function describes the value of a stock over time, the derivative tells us the instantaneous rate of return – in other words, how fast the stock price is rising or falling at that exact moment.
Derivatives apply to functions and describe the rate of change of a function with respect to its input. Put simply: when we differentiate a function, we obtain another function that tells us how the original changes at any point.
If a function increases at the same rate as its input, its derivative at that point equals 1.
If it grows faster or slower, the derivative is greater or less than 1.
If the function is constant, the derivative is 0.
If the function decreases as the input increases, the derivative is negative.
Some functions are not differentiable at certain (or even all) points. For those where derivatives do exist, we say the function is differentiable in that domain.
Derivatives appear across science and engineering:
In physics, to describe motion and velocity, or in electromagnetism to calculate current.
In chemistry, to model the kinetics of reactions.
In biology, to analyze population growth rates.
In engineering, to evaluate stress, flow, or optimization problems.
And of course, much more. In fact, derivatives are so essential that it’s hard to imagine modern science without them.
Students often ask: “What is the point of this? Will I ever use derivatives in real life?” There’s a story about Euclid, who dismissed a student from his geometry school for asking a similar question. His reasoning? Mathematics exists for its own sake, not just for practical use.
That said, if you ever struggled with derivatives or wish to dive deeper, you can always turn to eMatematika in Croatia, one of the first European platforms for online tutoring specialized in mathematics, physics and computer science (instrukcije iz matematike i fizike) and Croatian Matura exam online course (pripreme za maturu online).
Euclid’s student may not have been entirely wrong. Derivatives demonstrate how mathematics and physics are intertwined: without real-world problems, much of mathematics might not have been developed at all. Yet mathematics also stands on its own, a discipline valued for both practical applications and pure abstraction.
With this article, we hope to have presented derivatives in a way that is both engaging and (non)formal.
