Calculus is one of the most important fields of Mathematics. Calculus is a study of rates of change and motion, which we can see by the slope of a line or a curve. There are two major branches of calculus, Differential and Integral calculus, and they are inverses of each other. Integral calculus is used to find the areas under a curve, surface area or volume, and linear distance travel. Differential calculus (which concerns the derivative) mostly goes over the problem of finding the rate of change that is instantaneous, for example, the speed , velocity or an acceleration of an object. Differentiation is especially important in natural sciences, engineering and technology.
An example of differential calculus is if you wanted to find the velocity or the acceleration of an object, for example, a car. To find the velocity of a car, you would take the first derivative of a function (position at time t : dx/dt) and to find the acceleration you would take the second derivative of a function (dv/dt : change in velocity/change in time . This leads us to Newton’s law of motion, which is Force = Mass x Acceleration, where in this context, acceleration is the second derivative of a function.
Who was the person behind the development of calculus? Well, it wasn’t actually just one person. Sir Isaac Newton and Gottfried Wilhelm Leibniz were both credited with the development of calculus. Throughout their lives, they both argued on who came up with the idea first, both have accused each other of plagiarism. Those two weren’t the only ones who contributed to the discovery of Calculus. There have been many other known mathematician of that time that also helped with the development of calculus. For example, Rene Descartes indirectly helped create differential calculus by introducing variable magnitude.
Newton and Leibniz essentially created integral and differential calculus. They were both interested in objects that are in motion. However, they both looked at different aspects of this. Newton was more involved with the speed of a falling object and Leibniz with the slopes of curves to illustrate the rate of change. Although they both looked at different things, they both came up with the same results, hence the accusations of stealing the other’s ideas. However, combining both of their ideas, fundamental theorem of calculus was created, which links the concept integration to derivation.
It is hard to see the difference between the function and its derivative without having a visual presentation. In math, graphs are usually used to show what a function and its derivative look like. Any value of the first derivative at a given point is equal to the slope of the tangent to the graph of the function at that point. As we all know that in a graph, positive means increasing, so when the derivative is positive, the function must be increasing and when the derivative is negative, the function must be decreasing. When the value is zero at a point, the tangent is horizontal, and the function changes from increasing to decreasing, or from decreasing to increasing, depending on the value of the second derivative. The second derivative basically represents the curvature of the function. Since the first derivative shows the rate of change, the second derivative shows the rate of change of the rate of change. When the second derivative is positive, the function concave upwards and when the second derivative is negative, the function concave downwards.
To find a derivative of a function we have to make sure that the two x values are as close as possible so we can receive an accurate result. Derivative is defined by the limit of slope formulas as the x values become closer to each other. For example, we take a point which is on a curve, now we take another point that is closest to x, x+delta x. All we need to do now is plug this into the slope formula, one more thing, since we want the closest value to x, delta x has to be very small, so we find the derivative as delta x goes to 0; now we have the entire formula for derivative shown in the image.
Differential Calculus helped evolve Math in many ways. It is used in many different fields of science, such as, physics, biology, and engineering.