# What differential calculus and integral calculus. The differential

What is Calculus? The word “calculus” in the Latin language, means “rock” or “small pebble” where ancient people used small rocks and stones to do particular computations in mathematics.

Basically, Calculus is a branch of mathematics that studies the rate of change, motion, and area of a curve. It was discovered and developed in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. It has two (2) major branches that are related to each other. These are differential calculus and integral calculus. The differential tells us on how to get the rates of change of a slope or a curve while the integral talks about finding the area, volumes, lengths, and values, within the slope.

Calculus us the language of the engineers, economists, architects, and scientists. Their works have a great impact on our daily lives, from our cellphones, gadgets, in cooking, economy, weather forecasts, and constructions. With calculus, we have the ability to find the effects of the changing conditions on a system. And in by using calculus we can take advantage of it where we are able to change, to mode, and control particular systems in the modern world. But who is the first to discover calculus? According to the site Fitchburg State University, Newton and Leibniz’s calculus controversy, there was a large debate and argument about the discovery of calculus. Sir Isaac Newton made his discoveries in the late 1664-1666 but did not published it right away and was held back until they finally recognized and published it at 1693. Gottfried Wilhelm Leibniz, also made his discoveries right after Newton in the late 1672-1676 and got published in the year 1684 and 1686 before Newton. It raised a controversy because of the difference in the timeframe on when they discovered and published thus questioning Gottfried Wilhelm Leibniz for plagiarism of Newton’s work and ideas and inserted his own unique notation styles.

But eventually, the mathematical community realized and confirmed that Leibniz and Newton made their discoveries independently thus making them both the founder or the inventor of calculus. Newton and Leibniz both contributed a large compound in calculus. According to the “Story of Mathematics”, Isaac Newton has one of the greatest minds in history, he is a physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian. The problem that Newton was facing was that, it is easy to solve and compute for the average slope of a curve, it has no definite answer thus constantly varying, and there was no method to give the exact slope at any one individual point on the curve. Therefore, he discovered a derivative function that gives the slope or the rate of change of any individual point of the function. This type of process that calculates the slope or derivative of a function is called differential calculus or differentiation. Other term is “method of fluxions”.

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Newton was able to solve for the areas within the curve or tangent line that others can’t. Technically, if based on precise dates, Isaac Newton was the first to discover calculus but he didn’t publish his work after discovering because he feared that they would just make a mockery about his findings and his unconventional ideas. Leibniz on the other hand, also discovered calculus in his own manners. They both worked independently.

Leibniz was a German mathematician and a philosopher. Just like Newton, Leibniz  also created a similar theory of calculus. He discovered the infinitesimals. Infinitesimal calculus is a branch of mathematics that is based on differentiation, integrations, and the limits of a function. Including the numbers that are so small and impossible to measure. Leibniz was more than happy to publish his work unlike Newton. And so Europe first heard about calculus from Leibniz and not from Newton who published nothing until 1693.

Gottfried Wilhelm Leibniz was a member of the Royal Society in London, and when the Royal Society was asked to adjudicate between the rivals, Newton and Leibniz, they gave Newton the credit for being the first to discover calculus but to Leibniz, they gave him the credit of first publication. However, it was biased when Isaac Newton was still the president of the Royal Society resulting of the accused plagiarism of Leibniz. Though Leibniz was accused of plagiarism, his unique notations still gloriously succeeded than Newton’s confusing and clumsy notations. Eventually years after the death of Leibniz, the Royal Society finally realized that Leibniz was innocent even though they have almost similar concepts but very different where Newton relates to the physical world while Leibniz is more in the graphical concepts and numbers.           We may not be able to realize that even for us, ordinary people, we too are interacting with the principles of calculus. The professionals are not the only one dealing with basic concepts like the slopes, tangents, area, etc., Even in our day to day lives, there is a sense of calculus, if we see things in a close perspective.             First example is in traveling.

From your house to school. Calculus is about finding the rate of change, velocity, speed, and limit. If you are traveling with a jeep, and the distance between your house to school is about 30 kilometers. The speed limit is 50 kilometers an hour and you have at least an hour left before the bell rings. With the concept of calculus, you can calculate on how fast must the jeep will run. But let’s be realistic, a jeep can not run 30 kilometers an hour constantly.

The road has obstacles like other cars and stop lights that can affect the rate of change in motion of the jeep. There is no way a jeep can run 30 kilometers an hour without changing its velocity. But it can change its velocity, increasing in a certain point and decrease in other points. The jeep can run with continuous increasing speed but can never and can not reach 50 kilometers an hour.           Second example is what we do everyday, walking. You where about to go the comfort room across the hallway with a straight line. The distance is about 30 meters away.

You don’t know your acceleration and speed because you can’t measure them. And you wanted to know what is your velocity at a certain point, for example at the 18th meter. To get your velocity you need to  solve the displacement over time  elapsed. You can’t actually measure your velocity at only one point but you need two points.

The first displacement over time elapsed of the first point minus the second one. And as you decrease the distance between the point and get closer and closer to the 18th meter, you also get closer to the absolute velocity of your travel time.            Third example is gift wrapping every seasons and birthdays. In calculus, finding the best area is also one factor. Having a beautiful and memorable present for your loved ones is one of the factors that means you really care for them, but without the right boxes for their presents is a bummer. In making a box that has the biggest or largest area for your limited sized cardboard, consists with the three(3) dimensions which are the length, width, and height. There are several methods in solving for the largest area ,but the most effective method is the quadratic formula.

You can also use trial and error and having precise measurements in each dimensions, searching for the limit of the height, length, and width to get the best results.             In conclusion, calculus is one of the greatest and most important discovery that mankind has found. Most of us may not sense that we are using calculus in our everyday lives, but it is still applicable not only to professionals but even to us ordinary people. Regarding the founder of calculus, Sir Isaac Newton was credited to be the first founder of calculus and Gottfried Wilhelm Leibniz was credited for the first publication, but both of them simultaneously and independently discovered calculus.